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...s is equivalent . A Baer ring is necessarily a left and a right [[Rickart ring]]. * Tsit-Yuen Lam, "Lectures on Modules and Rings" Graduate Texts in Mathematics '''189''' Springer (2012) {{ISBN|1461205255}} {{ZBL|0911.16001}}

...[2]]] established that any [[Noetherian ring|Noetherian ring]] is a Lasker ring. ...R><TD valign="top">[3]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T

''$M$ over a commutative ring $R$'' .... For example, for a finite Abelian group$M$ regarded as a module over the ring of integers, $\mathrm{Supp}(M)$ consists of all prime ideals $(p)$, where $

...acobson ring. On the contrary, a local non-Artinian ring is not a Jacobson ring. ...f variables and the cardinality of the field $K$. A ring $A$ is a Jacobson ring if the space of maximal ideals of $A$ is quasi-homeomorphic to the spectrum

A topological ring is a [[Ring|ring]] $R$ that is a [[Topological space|topological space]], and such that the ...a topological ring. A direct product of topological rings is a topological ring in a natural way.

''unique factorisation domain'', ''Gaussian ring'' A ring with unique decomposition into factors. More precisely, a factorial ring $A$ is an [[Integral domain|integral domain]] in which one can find a syste

...if the localization $A_{\mathfrak{P}}$ is not an integrally-closed [[local ring]]. In geometrical terms this means that the conductor determines a [[closed <TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T

...$A$) is Noetherian if and only if $A/J$ is a [[Noetherian ring|Noetherian ring]], where $J$ is the [[nil radical]] of $A$. <TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) {{MR

...= R$, then $R$ is said to be integrally closed in $S$ (cf. also [[Integral ring]]). ...nd is closed in its field of fractions. In some of the literature a normal ring is also required to be an integral domain.

'''Algebra''' is a branch of [[mathematics]]. The term is used in combinations such as [[homological algebra]], [[comm ...r or vector algebra) over a field, over a skew-field or over a commutative ring. [[Associative algebra]]s (formerly described as "hypercomplex systems" )

$#C+1 = 15 : ~/encyclopedia/old_files/data/K055/K.0505930 Krull ring, ...valuations on the field of fractions (cf. [[Fractions, ring of|Fractions, ring of]]) $ K $

...0 = a$ for every $a \in S$. The most important classes of semi-rings are [[ring]]s and [[distributive lattice]]s. If there is a multiplicative [[unit eleme ...n-negative integers with the usual operations provide an example of a semi-ring that does not satisfy this condition.

...y the elements of the form $\sigma-1$, $\sigma\in G$, is nilpotent. In the ring of upper-triangular matrices over a field the matrices with 0's along the m ...in a [[Noetherian ring|Noetherian ring]]. In a left (or right) Noetherian ring every left (right) nil ideal is nilpotent.

...ural structure of a topological module over the completion $\hat R$ of the ring $R$. <TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) {{ZBL

A formal Dirichlet series over a ring $R$ is associated to a function $a$ from the positive integers to $R$ ...(so that $L(\delta,s)=1$) as multiplicative identity. An element of this ring is invertible if $a(1)$ is invertible in $R$. If $R$ is commutative, so is

An ideal $I$ in a ring which cannot be expressed as the intersection of two strictly larger ideals * {{Ref|a1}} D.G. Northcott, "Ideal Theory", Cambridge Tracts in Mathematics '''42''' Cambridge University Press {{ISBN|0-521-60483-4}} p.21

...fore that rings of sets are also closed under finite intersections. If the ring $\mathcal{A}$ contains $X$ then it is called an [[Algebra of sets|algebra o A $\sigma$-ring is a ring which is closed under countable unions, i.e. such that

$#C+1 = 33 : ~/encyclopedia/old_files/data/D030/D.0300550 Dedekind ring An associative-commutative ring $ R $

$#C+1 = 19 : ~/encyclopedia/old_files/data/C023/C.0203010 Coherent ring ...herent ring is defined similarly in terms of right ideals. A left coherent ring can also be defined by either of the following two equivalent conditions: 1

...multiplication]] and addition). Cf. also [[Boolean algebra]] and [[Boolean ring]] for the symmetric difference operation in an arbitrary Boolean algebra. ...D valign="top"> P. R. Halmos, ''Naive Set Theory'', Undergraduate Texts in Mathematics, Springer (1960) {{ISBN|0-387-90092-6}}</TD></TR>

...\Phi$ is the ring of integers, a free algebra over $\Phi$ is called a free ring (cf. [[Free associative algebra]]). Kurosh showed that a non-null subalgeb ...orial methods. Free groups, polynomials, and free algebras'', CMS Books in Mathematics '''19''' Springer (2004) {{ISBN|0-387-40562-3}} {{ZBL|1039.16024}}

of an integral commutative ring $ A $ of the [[field of fractions]] of the ring $ A $

...73 : ~/encyclopedia/old_files/data/I051/I.0501450 Integral extension of a ring of a commutative ring $ A $

A [[Linear topology|linear topology]] of a ring $ A $ of the ring $ A $

''on a ring $A$'' A topology on a ring for which there is a fundamental system of neighbourhoods of zero consistin

An ultrametric [[valuation]] $\Vert{\cdot}\Vert$ on a ring $R$ similarly satisfies the condition: $\Vert x+y \Vert \le \max\{\Vert x \ ...N. "An introduction to ultrametric summability theory". SpringerBriefs in Mathematics. Springer (2014) {{ISBN|978-81-322-1646-9}} {{ZBL|1284.40001}}

...a|commutative algebra]]. One says that Hensel's lemma is valid for a local ring $ A $ ...olution of the corresponding equation over its residue field. Thus, in the ring $ \mathbf Z _ {7} $

''of a commutative ring $R$'' ...ve analogue of primary decomposition. Every tertiary ideal of a Noetherian ring $R$ is primary if and only if $R$ satisfies the Artin–Rees condition: For

...o the ring $A$ the topological space $\text{Spec}\,A$ (the [[Spectrum of a ring|spectrum]] of $A$) consisting of all prime ideals of $A$. Then the assertio <TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T

...1 = 73 : ~/encyclopedia/old_files/data/D033/D.0303180 Discretely\AAhnormed ring, ''discrete valuation ring, discrete valuation domain''

...imes 2$ matrix ring $M_2(F)$. A quaternion algebra isomorphic to a matrix ring is termed ''split''; otherwise the quaternion algebra is a [[division algeb ...Lam, ''Introduction to Quadratic Forms over Fields'', Graduate Studies in Mathematics '''67''' , American Mathematical Society (2005) {{ISBN|0-8218-1095-2}} {{ZB

...ferences|[3]]]). The isolated prime ideals of an ideal $I$ of a polynomial ring over a field correspond to the irreducible components of the [[affine varie <TR><TD valign="top">[5]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T

$#C+1 = 81 : ~/encyclopedia/old_files/data/P074/P.0704740 Principal ideal ring An associative ring $ R $

...encyclopediaofmath.org/legacyimages/c/c025/c025920/c0259202.png" /> over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l ....encyclopediaofmath.org/legacyimages/c/c025/c025920/c02592031.png" /> be a ring with an identity, let <img align="absmiddle" border="0" src="https://www.en

of a ring $ R $ For an associative ring the following is an equivalent definition in terms of elements:

A transition from a commutative ring $ A $ to the ring of fractions (cf. [[Fractions, ring of|Fractions, ring of]]) $ A [ S ^ {-1} ] $,

The module of homomorphisms of a given module into the ground ring. More precisely, let $ M $ be a left module over a ring $ R $.

...t; for example, to the lattice of ideals in a [[Noetherian ring|Noetherian ring]]. ...R><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T

A [[module]] $M$ over a [[commutative ring]] $A$ The classes of isomorphic invertible modules form the Picard group of the ring $A$;

''polynomial ring'' ...depends only on a finite number of variables. A ring of polynomials over a ring $R$ is a (commutative) [[free algebra]] with an identity over $R$; the set

integers form a ring which contains the ring of integers. The ring of quotient field $\Q_p$ of the ring $\Z_p$ of $p$-adic integers. This field is

...enerators: Every mapping from the set of free generators into an arbitrary ring of the variety can be extended to a homomorphism. ...ities. A variety generated by a finite associative ring or by a finite Lie ring is a Specht variety. The question whether every variety of associative alge

Let $K$ be an algebraic number field with ring of integers $R$. A ''modulus'' is a formal product ...to 1 modulo $\mathfrak{p}^n$ in $R_{\mathfrak{p}}$ in the usual sense of ring theory. For a real place $\mathfrak{r}$ we define $x$ and $y$ to be congr

By the Pontryagin–Thom theorem, there is a ring spectrum $ MU $( ...ctrum of a ring]]) whose [[Homotopy|homotopy]] is isomorphic to the graded ring of bordism classes of closed smooth manifolds with a complex structure on t

''of a left module $M$ over a ring $R$'' ...do exist have been characterized [[#References|[a1]]] (cf. also [[Perfect ring]]).

An element of the semi-group $D$ of divisors (cf. [[Divisor|Divisor]]) of the ring $A$ of integers of an algebraic number field. The semi-group $D$ is a free ...with the absence of uniqueness of factorization into prime factors in the ring of integers of an algebraic number field. For every $a\in A$, the factoriza

more generally, a ring $ R $) over the ring of functions on $ M $.

...t to be a two-sided ideal that is invariant under all endomorphisms of the ring. The socle can be represented as a direct sum of simple modules. [[Complete <TR><TD valign="top">[a1]</TD> <TD valign="top"> L.H. Rowen, "Ring theory" , '''1''' , Acad. Press (1988) pp. §2.4</TD></TR>

...am409>Lam (2005) p.&nbsp;409</ref> For non-formally real fields, the Witt ring is torsion, so this agrees with the previous definition.<ref name=Lam410>La * Oleg T. Izhboldin, "Fields of u-Invariant 9", ''Annals of Mathematics'', 2 ser '''154''':3 (2001) pp.529–587 [http://www.jstor.org/stable/3062

$#C+1 = 98 : ~/encyclopedia/old_files/data/D031/D.0301210 Derivation in a ring of a ring $ R $

...aluation|valuation]] $v$ and if the Hensel lemma is true for the valuation ring of $v$, then $(K,v)$ is called a Henselian field and $v$ is called a Hensel See also [[Hensel ring|Hensel ring]].

$#C+1 = 93 : ~/encyclopedia/old_files/data/L060/L.0600190 Local ring A [[Commutative ring|commutative ring]] with a unit that has a unique maximal ideal. If $ A $

...ideals (cf. [[Divisorial ideal|Divisorial ideal]]) of a [[Krull ring|Krull ring]] $ A $ into irreducible factors. Thus, a factorial ring has trivial divisor class group.

...ication) and [[neutral element]]s $0$ and $1$ is called an idempotent semi-ring if the following basic properties are valid for all elements $a,b,c \in A$: ...References|[a3]]], [[#References|[a9]]]. The concept of an idempotent semi-ring is a basic concept in [[idempotent analysis]]. This concept has many applic

''over a ring $A$ in commuting variables $T_1,\ldots,T_N$'' The set $A[[T_1,\ldots,T_N]]$ of all formal power series forms a ring under these operations.

...or syzygy, between the $m_i$ is a set $(a_i)_{i\in I}$ of elements of the ring $A$ such that $\sum_{i\in I} a_i m_i = 0$. Thus there arises the module of ...course in commutative algebra and algebraic geometry'', Graduate Texts in Mathematics '''229''', Springer-Verlag (2005) {{ISBN|0-387-22232-4}} {{ZBL|1066.14001}}

The tensor algebra of a unitary module $V$ over a commutative associative ring $A$ with unit is the algebra $T(V)$ over $A$ whose underlying module has th ...s associative, but in general not commutative. Its unit is the unit of the ring $A = T^0(V)$. Any $A$-linear mapping of the module $V$ into an associative

$#C+1 = 32 : ~/encyclopedia/old_files/data/H110/H.1100300 Hopf ring A (graded) [[Ring|ring]] object in the [[Category|category]] of (graded) co-commutative co-algebra

<TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T over a filtered ring $ R $

[[Derivation in a ring|Derivation in a ring]]) of $k$ extends to an integral domain that is finitely generated as a ring over a field

the tensor product of forms induces on it the structure of a ring, which is known as the Witt–Grothendieck of $ k $[[#References|[7]]]. ...'s theorem see [[Witt decomposition|Witt decomposition]]; [[Witt ring|Witt ring]].

...is a [[group]] for Dedekind rings and only for such rings (cf. [[Dedekind ring]]). The [[invertible element]]s of the semi-group $\mathfrak A$ are said to <TR><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T

...r applying unexpected analogies and ideas borrowed from different areas of mathematics and theoretical physics [[#References|[a1]]], [[#References|[a2]]]. ...="top">[a2]</TD> <TD valign="top"> G.L. Litvinov, V.P. Maslov, "Idempotent mathematics: correspondence principle and applications" ''Russian J. Math. Phys.'' , '

...yimages/c/c025/c025970/c02597061.png" /> (i.e. the vector fields) over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l ...><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Multilinear algebra" , Addison-Wesley (197

...l A_1$ for each $B$ in some class of sets $B\in\mathcal A_2$ such that the ring generated by it is the whole of $\mathcal A_2$, then $f$ is measurable. The ...R><TD valign="top">[3]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from F

...heory obtained by tensoring oriented [[Cobordism|cobordism]] theory with a ring of modular forms in characteristic $ \neq 2 $. be oriented bordism theory with coefficient ring $ \Omega _ {*} ^ { { \mathop{\rm SO} } } $,

...is categorically dual to that of an injective envelope. If $R$ is a [[Ring|ring]], Bass proved that every left $R$-module has a projective cover if and onl ...[#References|[a4]]] gave two different solutions of the conjecture for any ring. One proof uses a result of El Bashir which shows that any morphism $F \rig

$#C+1 = 129 : ~/encyclopedia/old_files/data/W098/W.0908080 Witt ring ring of types of quadratic forms over $ k $''

...nt of the free non-associative algebra over $K$ (cf. [[Free algebra over a ring]]) by the ideal generated by all elements of the form $[ x, [y,z]] - [[x,y] ...orial methods. Free groups, polynomials, and free algebras'', CMS Books in Mathematics '''19''' Springer (2004) {{ISBN|0-387-40562-3}} {{ZBL|1039.16024}}

...nalysis" is understood in the wide meaning of the word, including discrete mathematics. ...re employed in the study of discrete structures. Thus, the two branches of mathematics merge to some extent.

mathematics, especially so with algebraic geometry and algebraic [[Commutative ring|Commutative ring]], see also

Here $R$ is a [[unital ring|ring with a unit element]] and equipped with an involutory [[anti-automorphism]] If $R$ is a commutative ring with identity, if $R_0 = \{r\in R : r^J = r\}$, and if the matrix of $\phi$

For an algebra, a ring or a semi-group $ A $, is isomorphic to the quotient ring or quotient algebra $ A/ \kappa $,

...belian group|Abelian group]] (which can be considered as a module over the ring $\mathbf Z$). In the non-Abelian case two concepts of the rank of a group a ...or of the system of columns (column rank). For matrices over a commutative ring with a unit these two concepts of rank coincide. For a matrix over a field

...rm on a field|Norm on a field]]) can be made into a logarithmically-valued ring if one passes in the groupoid of values from the multiplicative to the addi ...mic valuation to one with a non-Archimedean norm is also possible. If in a ring a non-Archimedean [[Real norm|real norm]] is given, then the corresponding

...integers, two polynomials (and, in general, two elements of a [[Euclidean ring]]) or the common measure of two intervals. It was described in geometrical <TR><TD valign="top">[b1]</TD> <TD valign="top"> John Stillwell. ''Mathematics and Its History'', 3rd revised and updated ed. Springer (2010). {{ISBN|978

...o called $\sigma$-ideal) of subsets of a given space $X$, i.e. a $\sigma$-ring $\mathcal{R}\subset \mathcal{P} (X)$ with the property that for every $E\i ...} (X)$ of subsets of $X$. Indeed observe that, if an hereditary $\sigma$-ring is also an [[Algebra of sets|algebra]], then it must contain $X$ and hence

[[Module|module]] $M$ into a (left) module $N$ over the same ring $A$, satisfying the conditions ''Elements of mathematics'', '''1''', Addison-Wesley (1973)

...sm of (left) modules with common ring of scalars (in this, the role of the ring is played by the domain of definition of the functors, and the functors the ...Mac Lane, ''Categories for the Working Mathematician'', Graduate Texts in Mathematics '''5''', Springer (1998) {{ISBN|0-387-98403-8}}</TD></TR>

ring homomorphism $\phi:A\otimes B \to C$ that associates with an element $x\oti |valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Algebra", ''Elements of mathematics'', '''1''', Springer (1988) pp. Chapts. 4–7 (Translated from French) {{MR

...ed an alternating matrix) of order $2n$ over a commutative-associative ring $A$ with a unit, then $\Pf X$ is the element of $A$ given by the formula |align="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Elements of mathematics", '''2. Linear and multilinear algebra''', Addison-Wesley (1973) pp. Chapt.

...odern algebra. Its principal task is to represent any [[Ideal|ideal]] of a ring (or of another algebraic system) as the intersection of a finite number of be a Noetherian ring, i.e. an associative ring with the maximum condition for ideals. If $ A $

...a left unitary $A$-module, $W$ is a right unitary $A$-module, and $A$ is a ring with a unit element, which is also regarded as an $(A,A)$-bimodule. If $V=W ...sometimes identified with $f$ and is called a bilinear form on $V$. If the ring $A$ is commutative, a bilinear form is a special case of a

A right order in a ring $ Q $ ...al right ring of fractions of $ R $ (see [[Fractions, ring of|Fractions, ring of]]).

This ring is local (cf. [[Local ring|Local ring]]). Its maximal ideal $ P $ ...R><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T

...groups. In particular, if $M$ is a free finite-dimensional module over the ring $\mathbf Z$ of integers, one speaks of crystallographic groups (cf. [[Cryst ...ms" A.D. Aleksandrov (ed.) A.N. Kolmogorov (ed.) M.A. Lavrent'ev (ed.) , ''Mathematics, its content, methods and meaning'' , '''3''' , Amer. Math. Soc. (1962) pp.

determine the same element of the [[Witt ring|Witt ring]] $ W(k) $ into the ring of integers $ \mathbf Z $.

over a commutative ring $ A $ ...ra of a module is employed in the theory of modules over a principal ideal ring [[#References|[5]]].

...tions of finite union and difference. An algebra can be characterized as a ring containing the set $X$. ...led Boolean $\sigma$-algebra or $\sigma$-field). Analogously one defines [[Ring of sets|$\sigma$-rings]] (also called Boolean $\sigma$-rings) as rings of s

...isomorphic to the field $\mathbb{Z}/p\mathbb{Z}$ of residue classes of the ring of integers modulo $p$. In any fixed [[Algebraic closure|algebraic closure] ...pha$ will be any root of any irreducible polynomial of degree $n$ from the ring $\mathrm{GF}(p)[X]$. The number of primitive elements of the extension $\ma

A polynomial $f$ with coefficients in a field or a commutative associative ring $K$ with a unit, which is a [[Symmetric function|symmetric function]] in it ...| V. V. Prasolov, "Polynomials", vol. 11 of "Algorithms and Computation in Mathematics", Springer (2004).

.... a [[semi-ring]] with idempotent addition). Let $A$ be an idempotent semi-ring with operations $\oplus$ (addition) and $\odot$ (multiplication) and neutra ...interesting numerical algorithm, then there is a good chance that its semi-ring analogues are important and interesting as well" , [[#References|[a9]]], [[

...[[Ring of polynomials|ring of polynomials]] is an associative-commutative ring without zero divisors (that is, a product of non-zero polynomials cannot be ...the ring of polynomials similar to that played by the prime numbers in the ring of integers. For example, the following theorem holds: If a product $PQ$ is

A [[totally ordered set|totally]] [[ordered ring]] which is a [[field]]. The classical example is the field of real numbers <TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics" , '''2. Algebra. Polynomials and fields. Ordered groups''' , Hermann (197

be an associative commutative algebra over a commutative ring $ R $( ...[Commutative algebra|Commutative algebra]]; [[Commutative ring|Commutative ring]]; [[Associative rings and algebras|Associative rings and algebras]]). A Po

...ign="top">[1]</TD> <TD valign="top"> O.Yu. Shmidt, , ''Selected works on mathematics'' , Moscow (1959) pp. 221–227 (In Russian)</TD></TR></table> (where $\mathbf Z_{p} $is the ring of integers of the $p$-adic completion $ \mathbf Q _ {p} $

...s the zero (i.e. the neutral element for addition) in the [[Witt ring|Witt ring]] of the field $ k $. cf. [[Witt ring|Witt ring]]).

...Abelian group|Abelian group]] with respect to addition (in particular, the ring has a zero element, denoted by 0, and a negative element $ - x $ in the ring.

...mum), that is, on the concept of an [[Idempotent semi-ring|idempotent semi-ring]]. In particular, idempotent analysis deals with functions taking values in ...|dynamic programming]], discrete-event systems, computer science, discrete mathematics, [[Mathematical logic|mathematical logic]], etc. (see, e.g., [[#References|

[[Abelian group]] with the distributive action of a ring. A module is a generalization of a (linear) [[ring]].

...xist for finding solutions of systems of such Diophantine equations in the ring of integers of any number field of finite degree over $ \mathbf Q $. be a system of polynomial equations having integral coefficients. The ring of all algebraic integers, $ {\widetilde{\mathbf Z} } $,

...e|Lie algebra, free]]) is freely generated (as a module over a commutative ring $ K $) ...lows: One has the direct sum decomposition (as a module over a commutative ring $ K $):

...and automata]]). In [[Mathematical logic|mathematical logic]] (i.e., meta-mathematics) one builds several classes of formal languages, of which first-order logic ...a ring is a two-sorted algebra with two universes, an Abelian group and a ring, and its language has two separate sorts of variables for the elements of t

* A field is quadratically closed if and only if its [[Witt–Grothendieck ring]] is isomorphic to $\mathbb{Z}$ under the dimension mapping.<ref name=Lam34 ...Lam, ''Introduction to Quadratic Forms over Fields'', Graduate Studies in Mathematics '''67''', American Mathematical Society (2005) {{ISBN|0-8218-1095-2}} {{ZB

is a commutative associative ring with a unit, cf. [[Associative rings and algebras|Associative rings and alg <TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley

...top">[2]</TD> <TD valign="top"> S.V. Yablonskii, "Introduction to discrete mathematics" , Moscow (1979) (In Russian) {{MR|0563327}} {{ZBL|}} </TD></TR></table> be homogeneous generators of the ring of invariants $ \mathbf R [ x _ {1}, \dots, x _ {n} ] ^ {G} $.

...ices that cannot even be imbedded in the lattice of ideals of a Noetherian ring [[#References|[3]]]). 2) The study of Noetherian modules over a multiplicat ...R><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T

...pper estimates are obtained for the height of the solutions, either in the ring of integers of a fixed algebraic number [[Field|field]] $ K $, or, more generally, in the ring of $ S $-

The branch of mathematics in which one studies algebraic operations (cf. [[Algebraic operation|Algebr ...algebra, its principal branches and its connection with other branches of mathematics.==

...oetherian [[Local ring|local ring]] (cf. also [[Noetherian ring|Noetherian ring]]) with [[Maximal ideal|maximal ideal]] $\mathfrak{m}$ and $d = \operatorna ...f $A$ is a Buchsbaum ring, then $A _ { \mathfrak{p} }$ is a Cohen–Macaulay ring with $\operatorname { dim } A _ { \mathfrak { p } } = \operatorname { dim }

...esults. The various family (relative) and equivariant versions of parts of mathematics are then often also important tools in the non-equivariant and non-family s ...n="top"> G. Carlsson, "Equivariant stable homotopy and Segal's Burnside ring conjecture" ''Ann. of Math.'' , '''120''' (1984) pp. 189–224</TD></TR>

...ver the field with $p$ elements, where $\mathcal{O}$ denotes the valuation ring of $v$. The $p$-rank is $1$ if and only if $p$ is a prime element and the r ...every choice of $a_1,...,a_n\in K$, $f(a_1,...,a_n)$ lies in the valuation ring $\mathcal{O}$ of $K$ whenever it is defined. For $K$ of $p$-rank $1$ this c

A non-zero positive [[Measure|measure]] $ \mu $ on the $ \sigma $-ring $ M $ of subsets $ E $ of a locally compact group $ G $ generated by the fa then there exists a positive measure $ \nu $ on the $ \sigma $-ring $ T $ of sets $ E \subseteq G / H = X $ that is generated by the family of

...concrete classes of rings. The fundamental object in number theory is the ring $ \mathbf Z $ ...the [[Hilbert polynomial|Hilbert polynomial]] for a homogeneous ideal in a ring of polynomials.

...ore generally) be considered on any ring with values in a linearly ordered ring [[#References|[4]]]. See also [[Valuation|Valuation]]. ...R><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T

...monic (with leading coefficient equal to 1), irreducible in the polynomial ring $k[x]$ and satisfying $f_\a(\a) = 0$; any constructed as the quotient ring $k[x]/fk[x]$. On the other hand, for any

...rious multiplicative [[Cobordism|cobordism]] invariants taking values in a ring of modular forms. The following is an attempt to present the simplest case is classified by a unique ring homomorphism

...ns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)-- in particular, t ...aracterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" r

...tely additive set function $\mu: \mathcal{C} \to [0, \infty]$ defined on a ring of sets, then $\mu$ is called (by some authors) regular, if ..."Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. {{MR|1158660}} {{ZBL|0804.2800}}

...branch of algebra, exhibiting many points of contact with other fields of mathematics and also with physics, mechanics, biology, and other sciences. The central ...is either special or is an Albert ring (a Jordan ring is called an Albert ring if its associative centre $Z$ consists of regular elements and if the algeb

Let $M$ be a [[module]] over a ring and $N$ a submodule. ...top"> Tsit-Yuen Lam, ''Lectures on Modules and Rings'', Graduate Texts in Mathematics '''189''', Springer (1999) {{ISBN|0-387-98428-3}}</TD></TR>

The Schubert calculus also refers to mathematics arising from the following class of enumerative geometric problems: Determi [[Chow ring|Chow ring]];

...a square matrix $A = (a_{ij})$ of order $n$ over a commutative associative ring $R$ with unit 1'' is ...number field), a ring of functions (especially a ring of polynomials) or a ring of integers.

...associative algebra structure. Then all derivations (cf. [[Derivation in a ring]]) of $V$ form a linear Lie algebra. If $V$ is a Lie algebra, then for a fi <TR><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from Fr

...some authors define the Borel sets as the smallest [[Ring of sets|$\sigma$-ring]] containing the compact sets, see {{Cite|Hal}}. Under suitable assumptions |valign="top"|{{Ref|Bou}}|| N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated fro

...bilinear form $\Phi$ on a left $K$-module $E$, where $K$ is a commutative ring (cf. ...symplectic group is called the symplectic group of $2m$ variables over the ring $K$ and is denoted by $\def\Sp{ {\rm Sp}}\Sp(m,K)$ or $\Sp_{2m}(K)$. The ma

...R><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from F be a ring of sets on some space $ \Omega $,

''over a ring $R$'' <TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from Fr

theory]] of a ring $ R $ ring spaces [[#References|[a4]]].

...iched by a unary [[relation symbol]] for being an element of the valuation ring (cf. [[Model theory of valued fields|Model theory of valued fields]]). ..."top"> A. Prestel, "Lectures on formally real fields" , ''Lecture Notes in Mathematics'' , '''1093''' , Springer (1984) {{MR|0769847}} {{ZBL|0548.12011}} </TD></T

...ring of polynomials in a finite number of variables over a field or over a ring of integers any ideal is generated by a finite number of elements (has a fi be a ring of polynomials over $ k $,

that is an element of a commutative ring. The weight of $ f $ .../TD> <TD valign="top"> V.N. Sachkov, "Combinatorial methods in discrete mathematics" , Moscow (1977) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD

...actical activities of mankind on the one hand, and the internal demands of mathematics on the other, have determined the development of the concept of a number. ...opment of the concept of number proceeded mainly along with the demands of mathematics itself. Negative numbers first appeared in Ancient China. Indian mathematic

...cycles in the co-invariant algebra, which is isomorphic to the cohomology ring of $\mathcal{F}_n$ [[#References|[a6]]]: ...nfty } = \cup \mathcal{S} _ { n }$. These form a basis for this polynomial ring, and every Schur polynomial is also a Schubert polynomial.

...generalization of the concept of a divisor of an element of a commutative ring. First introduced by E.E. Kummer ...ctorization, the elements of which are known as (integral) divisors of the ring $A$. The theory of divisors makes it possible to reduce a series of problem

This method of solving (1) over the ring of integers can be [[Euclidean ring|Euclidean ring]] or

over the ring of continuous functions on $ B $ ...etry; by now it has become a basic tool for studies in various branches of mathematics: differential and algebraic topology, the theory of linear connections, alg

The set of arithmetical functions forms a [[commutative ring]] with unity under the usual [[pointwise addition]] and the Dirichlet convo ...onvolutions" , ''The Theory of Arithmetic Functions'' , ''Lecture Notes in Mathematics'' , '''251''' , Springer (1972) pp. 247–271</TD></TR>

...{ - 1 , - 1 \}$), or for any semi-local ring or for any discrete valuation ring (in which case R.K. Dennis and M.R. Stein [[#References|[a1]]] have given a ...K$-theory, number theory, algebraic geometry, topology, and other parts of mathematics.

is a proper scheme over a complete local ring $ A $, be a Noetherian ring which is complete (and separated) with respect to the $ I $-

...damentals notions in general mathematics, and is used extensively, both in mathematics itself and in its applications. ...valign="top"> S.O. Shatunovskii, , ''Proc. 1-st All-Russian Congress of Mathematics Teachers'' , '''1''' (1913) pp. 276–281 (In Russian)</TD></TR><TR><TD

...f all derivations of $\frak g$ (cf. [[Derivation in a ring|Derivation in a ring]]). If $\frak g$ is a semi-simple Lie algebra (see [[Lie algebra, semi-simp ...<tr><td valign="top">[a1]</td> <td valign="top"> N. Bourbaki, "Elements of mathematics, Lie groups and Lie algebras" , Addison-Wesley (1975) pp. Chap. 2–3 (Tran

...a manifold. Coordinates are given different names in different branches of mathematics and physics; e.g. the coordinates of an element (vector) in a vector space ...a discovery that led to the concept of projective geometry over a division ring). Nevertheless, the idea still persists of a so to speak internal method of

see [[Rings and algebras|Rings and algebras]]; [[Operator ring|Operator ring]]). is the concept of a [[Module|module]] over an arbitrary ring. Some basic theorems of linear algebra cease being true if the vector space

...also displayed by algebraic integers. Thus, the algebraic integers form a ring; on the other hand, the real algebraic integers form an everywhere-dense se ...nother important difference between the ring of algebraic integers and the ring of rational integers. The concept of an irreducible integer (in analogy to

...m]]). This can be generalized to an Azumaya algebra $A$ over a commutative ring $R$ (cf. also ...uren, "Théorie de la descente et algèbres d'Azumaya", ''Lecture Notes in Mathematics'', '''389''', Springer (1974) {{MR|0417149}} {{ZBL|0284.13002}}

field of complex numbers, an arbitrary field, a ring of polynomials, the ring of integers, a ring of functions, or an arbitrary associative

...functors on various categories of algebraic objects (modules over a given ring, sheaves, etc.). ...in of application of homological algebra is the category of modules over a ring. Most of the results known for modules may be applied to abelian categories

Any Boolean algebra is a [[Boolean ring|Boolean ring]] with a unit element with respect to the operations of "addition" ( $ + any Boolean ring with a unit element can be considered as a Boolean algebra.

...roup consisting of matrices $(a_{ij})\in \GL_n(\Z_p)$ (where $\Z_p$ is the ring of $p$-adic integers), such that $a_{ij}\in p\Z_p$ for $i>j$, and let $N$ b |valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras", Addison-Wesley (1975) pp. Chapt. 4 (Translat

...s under discussion. The notion of a modulus figures in various branches of mathematics, although sometimes under other names — [[Absolute value|absolute value]] In more abstract situations it is natural to use an ordered semi-ring instead of $ \mathbf R ^ {+} $(

...her with a unary [[relation symbol]] for being an element of the valuation ring, or a binary relation symbol for valuation divisibility $v(x)\leq v(y)$ (cf ...t in the proof that Hilbert's 10th problem has a positive solution for the ring of all algebraic integers (see [[Algebraic Diophantine equations|Algebraic

where $a_{ij}$ are elements from a commutative ring and summation is over all one-to-one mappings $\sigma$ from $\{1,\ldots,m\} <TR><TD valign="top">[1]</TD> <TD valign="top"> H.J. Ryser, "Combinatorial mathematics" , Wiley &amp; Math. Assoc. Amer. (1963) {{ZBL|0112.24806}}</TD></TR>

...f\s{ {\sigma}}$ in a basis $(v) = \{v_1,\dots,v_n\}$ is the element of the ring $A$ equal to ...a fixed basis of a free $A$-module $E$ of finite rank over the commutative ring $A$ (with a unit element). If $(w) = \{w_1,\dots,w_n\}$ is another basis in

The endomorphism ring of a [[tilting module]] over a finite-dimensional hereditary algebra (cf. a ...Ringel, "Tame algebras and integral quadratic forms" , ''Lecture Notes in Mathematics'' , '''1099''' , Springer (1984) {{MR|0774589}} {{ZBL|0546.16013}} </td></t

...ant in applications. Every Tikhonov space is uniquely characterized by the ring of all continuous real-valued functions on it in the topology of pointwise ...character of these concepts and the results related to them, important for mathematics as a whole, makes their investigation most natural and transparent within t

be the ring of integers of an algebraic number [[Field|field]] $ F $( ...K-theory" H. Bass (ed.) , ''Algebraic K-theory II'' , ''Lecture Notes in Mathematics'' , '''342''' , Springer (1973) pp. 489–501</TD></TR>

...of problems, a broad range of methods and strong links with many fields of mathematics, both properly algebraic (primarily the theory of groups and rings) and oth ...pings that the links between the theory of semi-groups and other fields of mathematics have been realized. In this framework, semi-groups appear very frequently a

A collection $\mathcal{P}$ of subsets of $X$ is called a semi-ring of sets if A collection $ \mathcal{R}$ of subsets of $X$ is called a ring of sets if

...uct of two unitary modules $V_1$ and $V_2$ over an associative commutative ring $A$ with a unit is the $A$-module $V_1 \tensor_A V_2$ together with an $A$- ...or product of two algebras $C_1$ and $C_2$ over an associative commutative ring $A$ with a unit is the algebra $C_1 \tensor_A C_2$ over $A$ which is obtain

...eory. In a wide sense, the term "K-theory" is used to denote the branch of mathematics that includes algebraic $ K $-theory and topological $ K $-theory, and ...of $ K $-theory many problems both in topology and in other branches of mathematics were reconsidered. The most important results in $ K $-theory were connec

.../TD> <TD valign="top"> P.S. Urysohn, "Works on topology and other areas of mathematics" , '''1–2''' , Moscow-Leningrad (1951) (In Russian)</TD></TR></table> ==Dimension of an associative ring.==

''combinatorial mathematics, combinatorics'' The branch of mathematics devoted to the solution of problems of choosing and arranging the elements

''of a Lie algebra $ \mathfrak{g} $ over a commutative ring $ \mathbb{k} $ with a unit element'' N. Bourbaki, “Lie groups and Lie algebras”, ''Elements of mathematics'', Hermann (1975), pp. Chapts. 1–3. (Translated from French)</TD></TR>

...ear form|sesquilinear form]] $f$ on a right $K$-module $E$, where $K$ is a ring; here $f$ and $E$ (and sometimes $K$ as well) usually satisfy extra conditi In the more general situation when $E$ is a module over a ring $K$ the results on classical groups are not so exhaustive (see

ring generated by the cylinder subsets of $ B $( ...p"> H.H. Kuo, "Gaussian measures in Banach spaces" , ''Lecture Notes in Mathematics'' , '''463''' , Springer (1975)</TD></TR></table>

...and the notion of a [[Functor|functor]] related to it for many branches of mathematics. Many concepts and results in mathematics turn out to be dual to others from a category-theoretic point of view: inje

...valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , ''Lecture Notes in Mathematics'' , '''5''' , Springer (1994) (Edition: Fifth) {{MR|1324577}} {{ZBL|0812.12

be the [[field of fractions]] of a regular discretely-normed ring $ O $ Representable functors occur in many branches of mathematics besides algebraic geometry. S. MacLane [[#References|[a1]]] traces their fi

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al good-humoured grammar school senior teacher in mathematics,

...re). If a complex is shellable then its [[face ring]] is [[Cohen–Macaulay ring|Cohen–Macaulay]]. ...er, Bernd Sturmfels, "Combinatorial commutative algebra" Graduate Texts in Mathematics '''227''' Springer (2005) {{ISBN|0-387-23707-0}} {{ZBL|1090.13001}}</TD></T

...Abelian groups serve as a constant source of examples in various fields of mathematics.

...ng|Integral ring]]; [[Integral extension of a ring|Integral extension of a ring]]). If $ A $ ...R><TD valign="top">[4]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T

...nd algebras|Associative rings and algebras]]; [[Noetherian ring|Noetherian ring]]). A.M. Sinclair and A.W. Tullo showed [[#References|[a29]]] that a comple ...Neher, "Jordan triple systems by the grid approach" , ''Lecture Notes in Mathematics'' , '''1280''' , Springer (1980)</td></tr><tr><td valign="top">[a25]</td>

...anches of number theory, abstract algebra, combinatorics, and elsewhere in mathematics, it is sometimes possible and convenient to formulate a variety of enumerat ...true for the multiplicative semi-group $G_K$ of all non-zero ideals in the ring $R=R(K)$ of all algebraic integers in a given

...power-series completions of $ A $ and $ L $, i.e., $ \widehat{A} $ is the ring of power series in the associative but non-commutative variables $ u $ and N. Bourbaki, “Elements of mathematics. Lie groups and Lie algebras”, Addison-Wesley (1975). (Translated from Fr

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al political circumstances, Christiaan Huygens turned to science and mathematics.

the valuation ring of $ K $ ...t, "A compactification of the Bruhat–Tits building" , ''Lecture Notes in Mathematics'' , '''1619''' , Springer (1996)</TD></TR><TR><TD valign="top">[a3]</TD> <

...thematical structures of the same type. Correspondences are widely used in mathematics and also in various applied disciplines, such as theoretical programming, g Thus one has group correspondences, module correspondences, ring correspondences, and others. Such correspondences often have useful descrip

...hris, Y. Moschovakis (ed.) , ''Cabal Seminar '76-77'' , ''Lecture Notes in Mathematics'' , '''689''' , Springer (1978) pp. 193–208</TD></TR><TR><TD valign="to

...{p} ) $ of invertible matrices of order $ n $ with coefficients in the ring $ \mathbf Z _{p} $ of $ p $ -adic integers (with the topology induced ...><TD valign="top">[a3]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from Fr

...singular one-dimensional scheme (i.e., $ A $ is a [[Dedekind ring|Dedekind ring]]), then any algebraic vector bundle is the direct sum of a trivial and a l ...theory”, D. Quillen (ed.) G. Segal (ed.) S. Tsou (ed.), ''The Interface of Mathematics and Particle Physics'', Oxford Univ. Press (1990). {{MR|1103130}} {{ZBL|}}<

a unitary $k$-module $L$ over a commutative ring $k$ with a unit that is endowed with a bilinear mapping $(x,y)\mapsto [x,y] Lie algebras appeared in mathematics at the end of the 19th century in connection with the study of Lie groups (

Depending on the branch of mathematics, some spaces of mappings turn out to be especially important. Among the cen ...trary [[Tikhonov space]]s in direct relation to the study of [[topological ring]]s. More precisely, two Tikhonov spaces $X,Y$ are homeomorphic if and only

...pen subset $U\subseteq X$ an Abelian group $F(U)$ (a ring, a module over a ring, etc.) and to every pair of open sets $V\subseteq U$ a homomorphism $F_V^U: $\G_c(X,\cF)$) be the group (ring, module, etc.) of all sections of

...R><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley ...<TD valign="top">[a2]</TD> <TD valign="top"> F.D. Veldkamp, "Projective ring planes and their homomorphisms" R. Kaya (ed.) et al. (ed.) , ''Rings and

In classical mathematics, an orientation is the choice of an equivalence class of coordinate systems ...}^0(S^0))^*$, where $(\cdot)^*$ is the group of invertible elements of the ring $(\cdot)$.

.../TD> <TD valign="top"> V.N. Sachkov, "Combinatorial methods in discrete mathematics" , Moscow (1977) (In Russian)</TD></TR> ...ra of a partially ordered set $(X,\le)$ may be defined over a general base ring.

over a ring $ K $, ...x variable, which is a discrete Abelian group (and hence a module over the ring $ \mathbf Z $),

are turned into rings (cf. [[Ring|Ring]]) isomorphic via $ r _ {ij } \mapsto r _ {ik } = ( a _ {i} + a _ {k} ) can be embedded into the multiplicative semi-group of the coordinate ring of a suitable frame in some $ 5 $-

The vector spaces most often employed in mathematics and in its applications are those over the field $C$ of [[complex number]]s ...f a vector space is a special case of the concept of a [[module]] over a [[ring]] — a vector space is a [[unitary module]] over a field. A unitary module

...eplaced by Azumaya algebras {{Cite|Gr}}. An algebra $A$ over a commutative ring $R$ is an Azumaya algebra if it is finitely generated and central over $R$ |valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra", '''1''', Addison-Wesley (1

...r the multiplicative semi-group $G_K$ of all non-zero ideals in the [[Ring|ring]] $R = R(K)$ of all algebraic integers in a given [[Algebraic number|algebr ...implest special case of the semi-group $G_R$ of all non-zero ideals in the ring $R = R(K)$ of all integral functions in an algebraic function field $K$ in

The branch of mathematics in which one studies such properties of geometrical figures (in a wider sen ...extent the use of analytical tools, such as the construction of a homology ring by way of differential forms (skew-symmetric tensors) and differential oper

Now let $F$ be a [[Global field|global field]], $A$ its adèle ring (cf. also [[Adèle|Adèle]]), and let $V$ and $W$ be as before. Then one ma ...cquet, R.P. Langlands, "Automorphic forms on $GL(2)$" , ''Lecture Notes in Mathematics'' , '''114''' , Springer (1970)</TD></TR><TR><TD valign="top">[a5]</TD> <TD

which is a ring of sets, i.e. contains the intersection and the union of any two elements i The space of maximal centred systems of a normal base-ring with the standard given base of closed sets on it is a Hausdorff compactifi

...of complex integers, it became clear that the study of an arbitrary [[Ring|ring]] must begin with the construction of a divisibility theory in it. ...TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> , ''The history of mathematics from Antiquity to the beginning of the XIX-th century'' , '''1–3''' , Mos

over an associative commutative ring with a unit. The stated examples and properties of tensors are transferred, ...R><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley

induces a transformation (automorphism) of the ring of all polynomials $ k [ x _{1} \dots x _{n} ] $ ...g being algebraic consequences of them (the relations form an ideal in the ring of polynomials in the variables $ t _{1} \dots t _{m} $ ,

.../TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top"> A.G. Kurosh, "Ring-theoretical problems connected with Burnside's problem for periodic groups"

...re derived equivalent) if and only if $R ^ { \prime }$ is the endomorphism ring of a tilting complex $T \in K ^ { b } ( P _ { \Lambda } )$. ...''Proc. Ottawa Conf. on Representation Theory, 1979'' , ''Lecture Notes in Mathematics'' , '''832''' , Springer (1980) pp. 103–169 {{MR|607151}} {{ZBL|}} </td><

with $\sigma = \pm 1$. It arose in applied mathematics [[#References|[a4]]] but was soon recognized to be related to problems of [ ...associated to the algebro-geometric solutions is the spectral curve of the ring of differential operators in $x$ that commute with $\mathcal{L}$ and the ti

...of a mathematical system (a structure) has become a fundamental concept in mathematics. ...nature of their elements was first formulated as an independent branch of mathematics with the appearance of the book Abstract group theory by O.Yu. Shmidt (1916

...ldots , d$, the derivative $( d / d z ) f _ { i }$ of $f_i$ belongs to the ring $K [ f _ { 1 } , \ldots , f _ { d } ]$. Then the set of $w \in \mathbf{C}$ ...miny, 1997)" Y.V. Nesterenko (ed.) P. Philippon (ed.) , ''Lecture Notes in Mathematics'' , '''1752''' , Springer (2001)</td></tr><tr><td valign="top">[a7]</td> <t

...on of matrix polynomials, i.e., the study of the division structure of the ring of $(m\times m)$-matrices with polynomial entries, is a quite different mat ...oGr}}||valign="top"| D.M. Young, R.T. Gregory, "A survey of numerical mathematics", '''II''', Dover (1988) {{MR|1102901}} {{ZBL|0732.65003}}

1) $P _ { L } ( v , z )$ is an element of the Laurent polynomial ring $\mathbf{Z} [ v ^ { \pm 1 } , z ^ { \pm 1 } ]$. Furthermore, $( v z ) ^ { \ ...iant). More generally, consider $P _ { L } ( v , z )$ as an element of the ring $R$ that is the subring of $\mathbf{Z} [ v ^ { \pm 1 } , z ^ { \pm 1 } ]$ g

...non-linear, differential, integral, etc. For example, consider the normed ring $ [ X] = [ X \rightarrow X] $ ...TD valign="top">[7]</TD> <TD valign="top"> , ''Encyclopaedia of elementary mathematics'' , '''3. Functions and limits''' , Moscow-Leningrad (1952) (In Russian)<

...dean geometry is not the only possible one, and that it is advantageous in mathematics to develop other, non-Euclidean, geometries, irrespective of their relation ...in Hodge theory, yields information about the structure of the cohomology ring $ H ^ {*} M $

...op">[a2]</td> <td valign="top"> T.H. Brylawski, "The Tutte–Grothendieck ring" ''Algebra Univ.'' , '''2''' (1972) pp. 375–388</td></tr> ...ry" A. Barlotti (ed.) , ''Matroid Theory and its Applications Proc. Third Mathematics Summer Center C.I.M.E.'' , Liguori, Naples (1980) pp. 125–275</td></tr>

...and the concept of an ideal has now become fundamental in many branches of mathematics. ...K = k ( \theta ) $ or does it split in a product of prime ideals of the ring of algebraic integers of $ K $ ? In the latter case: By which law does it

...46420/h04642097.png" /> is canonically homeomorphic to the spectrum of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l ...cyclopediaofmath.org/legacyimages/h/h046/h046420/h046420233.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l

One of the basic concepts in mathematics. Let two sets $X$ and $Y$ be given and suppose that to each element $x\in X In ancient mathematics the idea of functional dependence was not expressed explicitly and was not

...of equations; thus, they are also known as indefinite equations. In modern mathematics the concept of a Diophantine equation is also applied to algebraic equation ...ucceeded in solving systems of equations with two unknowns. This branch of mathematics flourished to the greatest extent in Ancient Greece. The principal source i

...is fundamental in linear algebra, plays a role in very diverse branches of mathematics and physics, above all in analysis and its applications. ...l05934014.png" />). However, it was rooted in the previous developments of mathematics, which had accumulated (beginning with the linear function <img align="absm

...atics. In conjunction with algebra, topology forms a general foundation of mathematics, and promotes its unity. ...logy has to deal were described under the influence of various branches of mathematics in response to their very dissimilar requirements. This explains the a prio

...lgebras and their representations are deeply related to many directions in mathematics and physics, in particular, the representation theory of the Fischer–Grie ...$V$ (actually, this definition works over $\bf Z$ or any other commutative ring) equipped with an element $\mathbf{1}$, linear operators $D^{( i )}$ on $V$

The area of mathematics whose main object of study is the index of operators (cf. also [[Index of a ...call that the representation ring of a compact group $G$ is defined as the ring of formal linear combinations with integer coefficients of equivalence clas

be either the ring of integers $ \mathbf Z $ or the ring $ F[ \lambda ] $

...isolation of "reduced" forms in each class of quadratic forms over a given ring $ R $, ...in, M.I. Shtorgin, "The types of Bravais lattices" , ''Current problems in mathematics'' , '''2''' , Moscow (1973) pp. 119–254 (In Russian) {{MR|0412947}} {{ZBL

...eric fiber, e.g., sections of the vector bundle form a [[module]] over the ring of "scalar" functions. ...th notes by S. Ramanan. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, '''20''', 1965. Reprinted by Springer-Verlag, Berlin, 1986.

...Consistency]], [[Proof theory]], and [[Unsolvability]]. For the history of mathematics associated with this problem, see [[Hilbert 2nd problem]] and [[Hilbert pro ...is the most famous and important of the yet (1998) unsolved conjectures in mathematics. Its algebraic-geometric analogue, the Weil conjectures, were settled by P.

the ring of integral Laurent polynomials in $ \mu $ ...der invariants|Alexander invariants]]) an ascending chain of ideals of the ring $ \Lambda _ \mu $(

...ht vector spaces over a (not necessarily commutative) topological division ring. Sometimes a topological vector space $ E $ ...R><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from Fren

...e majority of functions which are encountered in the principal problems of mathematics and its applications to science and technology. Secondly, the class of anal ...ial class of polynomials in the variable $z_n$, with coefficients from the ring of holomorphic functions in the other variables $(z_1, \ldots, z_{n-1})$. T

[[Gorenstein ring|Gorenstein ring]]). <TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French