...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}}
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...[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
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''$M$ 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
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...$A$) 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
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...= 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" )
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$#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
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...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