...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}}
529 bytes (77 words) - 14:22, 12 November 2023
...[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
1 KB (157 words) - 11:43, 29 June 2014
''$M$ 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 $
918 bytes (149 words) - 20:46, 18 October 2014
...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
3 KB (415 words) - 20:32, 19 January 2016
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.
3 KB (410 words) - 21:45, 23 July 2012
''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
3 KB (480 words) - 21:45, 3 January 2021
...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
1 KB (161 words) - 08:20, 30 November 2014
...$A$) 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
1 KB (193 words) - 17:12, 29 December 2020
...= 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.
2 KB (305 words) - 16:09, 11 September 2016
'''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" )
788 bytes (106 words) - 11:22, 9 April 2017
$#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 $
3 KB (424 words) - 22:15, 5 June 2020
...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.
2 KB (371 words) - 05:54, 15 April 2023
...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.
3 KB (470 words) - 23:58, 24 November 2018
...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
1 KB (204 words) - 18:09, 17 April 2017
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
2 KB (358 words) - 17:25, 11 November 2023
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
332 bytes (55 words) - 18:11, 14 November 2023
...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
2 KB (231 words) - 09:47, 16 August 2013
$#C+1 = 33 : ~/encyclopedia/old_files/data/D030/D.0300550 Dedekind ring
An associative-commutative ring $ R $
4 KB (637 words) - 17:32, 5 June 2020
$#C+1 = 19 : ~/encyclopedia/old_files/data/C023/C.0203010 Coherent ring
...herent ring is defined similarly in terms of right ideals. A left coherent ring can also be defined by either of the following two equivalent conditions: 1
3 KB (479 words) - 17:45, 4 June 2020
...multiplication]] and addition). Cf. also [[Boolean algebra]] and [[Boolean ring]] for the symmetric difference operation in an arbitrary Boolean algebra.
...D valign="top"> P. R. Halmos, ''Naive Set Theory'', Undergraduate Texts in Mathematics, Springer (1960) {{ISBN|0-387-90092-6}}</TD></TR>
2 KB (273 words) - 08:47, 29 April 2023
...\Phi$ 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}}
1 KB (163 words) - 08:20, 12 November 2023
of an integral commutative ring $ A $
of the [[field of fractions]] of the ring $ A $
3 KB (406 words) - 21:43, 3 January 2021
...73 : ~/encyclopedia/old_files/data/I051/I.0501450 Integral extension of a ring
of a commutative ring $ A $
3 KB (562 words) - 22:12, 5 June 2020
A [[Linear topology|linear topology]] of a ring $ A $
of the ring $ A $
3 KB (559 words) - 16:09, 1 April 2020
''on a ring $A$''
A topology on a ring for which there is a fundamental system of neighbourhoods of zero consistin
2 KB (315 words) - 19:21, 14 August 2014
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}}
881 bytes (127 words) - 17:38, 13 October 2023
...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} $
3 KB (386 words) - 22:10, 5 June 2020
''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
3 KB (455 words) - 19:51, 5 October 2017
...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
2 KB (320 words) - 08:07, 1 January 2017
...1 = 73 : ~/encyclopedia/old_files/data/D033/D.0303180 Discretely\AAhnormed ring,
''discrete valuation ring, discrete valuation domain''
5 KB (800 words) - 19:36, 5 June 2020
...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
1 KB (180 words) - 20:49, 5 December 2023
...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
2 KB (386 words) - 20:06, 5 October 2017
$#C+1 = 81 : ~/encyclopedia/old_files/data/P074/P.0704740 Principal ideal ring
An associative ring $ R $
5 KB (880 words) - 19:00, 9 January 2024
...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
6 KB (880 words) - 17:29, 7 February 2011
of a ring $ R $
For an associative ring the following is an equivalent definition in terms of elements:
2 KB (393 words) - 08:07, 6 June 2020
A transition from a commutative ring $ A $
to the ring of fractions (cf. [[Fractions, ring of|Fractions, ring of]]) $ A [ S ^ {-1} ] $,
4 KB (713 words) - 21:35, 4 January 2021
The module of homomorphisms of a given module into the ground ring. More precisely, let $ M $
be a left module over a ring $ R $.
3 KB (435 words) - 16:09, 1 April 2020
...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
1 KB (217 words) - 08:34, 1 August 2014
A [[module]] $M$ over a [[commutative ring]] $A$
The classes of isomorphic invertible modules form the Picard group of the ring $A$;
2 KB (308 words) - 18:37, 11 April 2023
''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
6 KB (1,011 words) - 06:01, 30 September 2023
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
6 KB (1,089 words) - 20:29, 9 April 2017
...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
4 KB (701 words) - 17:01, 23 November 2023
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
3 KB (527 words) - 17:00, 23 November 2023
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
4 KB (633 words) - 08:47, 26 March 2023
''of a left module $M$ over a ring $R$''
...do exist have been characterized [[#References|[a1]]] (cf. also [[Perfect ring]]).
2 KB (368 words) - 16:20, 18 November 2023
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
5 KB (853 words) - 12:16, 22 August 2014
more generally, a ring $ R $)
over the ring of functions on $ M $.
2 KB (358 words) - 08:12, 6 June 2020
...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>
2 KB (371 words) - 15:04, 19 November 2023
...am409>Lam (2005) p. 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
3 KB (488 words) - 20:22, 15 November 2023
$#C+1 = 98 : ~/encyclopedia/old_files/data/D031/D.0301210 Derivation in a ring
of a ring $ R $
6 KB (945 words) - 17:32, 5 June 2020