...ic forms and are used in studies of residual approximability of arithmetic groups relative to conjugacy (see [[#References|[2]]]).
...n="top"> A.S. Rapinchuk, "Platonov's conjecture on the genus in arithmetic groups" ''Dokl. Akad. Nauk Bel.SSR'' , '''25''' : 2 (1981) pp. 101–104; 187 (In
2 KB (333 words) - 20:58, 29 November 2014
$#C+1 = 91 : ~/encyclopedia/old_files/data/Q076/Q.0706500 Quasi\AAhFrobenius ring,
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7 KB (1,087 words) - 08:09, 6 June 2020
....; it is usually regarded as an object of a certain [[Category|category]], and as a rule, the morphisms (cf. [[Morphism|Morphism]]) in this category are t
...ain proper sub-semi-groups of $\operatorname{End}X$ (for example, the semi-groups of homeomorphic transformations of a topological space).
3 KB (388 words) - 17:46, 25 October 2014
$#C+1 = 62 : ~/encyclopedia/old_files/data/O070/O.0700140 Ordered ring,
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7 KB (1,019 words) - 08:04, 6 June 2020
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The coincidence of the direct sum and the direct product in the case of a finite number of terms follows immediat
4 KB (680 words) - 19:35, 5 June 2020
...enerators: Every mapping from the set of free generators into an arbitrary ring of the variety can be extended to a homomorphism.
...bra with a finite number of generators over a field of characteristic zero and if $M_2\not\subseteq\mathfrak M$, then $\mathfrak M$ is a Specht variety.
4 KB (701 words) - 17:01, 23 November 2023
...n of a valued field]]; [[Ramification theory of valued fields|Ramification theory of valued fields]].
...general valuation theory and in the [[Model theory of valued fields|model theory of valued fields]], being Henselian has turned out to be more appropriate t
3 KB (405 words) - 19:33, 28 April 2014
...ldots,m$, are square matrices, the remaining places being filled by zeros, and each block $d_i(G)$ is an [[irreducible matrix group]]. In the language of
...<TD valign="top">[1]</TD> <TD valign="top"> Yu.I. Merzlyakov, "Rational groups" , Moscow (1987) (In Russian)</TD></TR>
2 KB (303 words) - 16:55, 10 April 2016
...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
...ule is complemented if and only if it is completely reducible and hence if and only if it coincides with its socle. The socle of $M$ can also be defined a
2 KB (371 words) - 15:04, 19 November 2023
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...concerning the relationship between direct decompositions of a group or a ring. The lattice-theoretical version of the result is known as Ore's theorem (s
5 KB (786 words) - 22:15, 5 June 2020
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whose additive group can be represented in the form of a (weak) direct sum of groups $ A _ {i} $,
4 KB (576 words) - 19:42, 5 June 2020
...The concept of a character of a group was originally introduced for finite groups $G$ with $A = T$ (in this case every character $G → ℂ^*$ takes values
...stood to mean characters of any of its finite-dimensional representations (and even to mean the representations themselves).
3 KB (438 words) - 13:36, 23 July 2015
...en we may take $Y_\lambda = \{ y_\lambda \}$ for all $\lambda \in \Lambda$ and then the restricted direct product becomes the [[direct sum]].
...ocally compact. This is the construction of the topology on the [[Idèle]] and [[Adele group]]s of a [[global field]].
2 KB (264 words) - 20:21, 20 November 2023
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...ject of an algebraic structure. The concept of an ideal first arose in the theory of rings. The name ideal derives from the concept of an [[Ideal number|idea
13 KB (2,163 words) - 22:11, 5 June 2020
$#C+1 = 98 : ~/encyclopedia/old_files/data/D031/D.0301210 Derivation in a ring
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6 KB (945 words) - 17:32, 5 June 2020
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is the ring of integers in the imaginary quadratic number field $ \mathbf Q ( \sqrt {
5 KB (775 words) - 07:59, 26 March 2023
...he characteristic property of an imbedding of a group (ring) into a group (ring) as a normal subgroup (ideal). Let $\mathfrak{K}$ be a [[Category|category]
...$H$. However, in an additive category the concepts of normal monomorphism and regular monomorphism coincide.
2 KB (314 words) - 02:26, 14 January 2017
the completion of $k$ with respect to $\nu\in V$, and $O_\nu$ is the
ring of integer elements in $k_\nu$. The adèle group of an algebraic
6 KB (1,071 words) - 06:53, 28 December 2021
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...igma$ of $R$ extends uniquely to an automorphism $\hat { \sigma }$ of $Q$, and one says that $\sigma$ is $X$-inner if $\hat { \sigma }$ is inner on $Q$ (c
9 KB (1,402 words) - 13:37, 25 November 2023
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A characteristic class defining a ring homomorphism $ \mathop{\rm ch} : K ( X) \rightarrow H ^ {**} ( X ; \m
5 KB (632 words) - 11:51, 21 March 2022