This page is a copy of the article [[Stability theorems in algebraic K-theory]] in order to test [[User:Maximilian_Janisch/latexlist|automatic LaTeXifica
...extensions of the ground ring $R$ (see [[Algebraic K-theory|Algebraic $K$-theory]]).
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...valid. There exists a group $K(C)$, called the Grothendieck group of $C$, and an additive mapping $k:\mathrm{Ob}(C)\to K(C)$, known as the universal mapp
...nerators — to each object $L \in C$, there corresponds a generator $[L]$ — and by the relations $[L]-[N]-[M]=0$ for each exact sequence $0 \to L \to M \to
4 KB (701 words) - 06:11, 26 March 2023
...t is hard to say whether field theory, the theory of finite groups and the theory of finite-dimensional Lie algebras should be regarded as general algebra.
...t developed theories are those of partially ordered and topological groups and rings.
5 KB (725 words) - 21:52, 30 March 2012
A pre-$\lambda$-ring is a commutative ring $R$ with identity element $1$ and a set of mappings $\lambda^n : R \rightarrow R$, $n = 0,1,2,\ldots$ such th
...ng exterior powers. For instance, for $M = \text{pt}$, $K(M) = \mathbf{Z}$ and the $\lambda$-structure is given by $\lambda^n(m) = \binom{m}{n}$ (binomial
10 KB (1,721 words) - 07:44, 23 March 2016
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...n which is an anti-isomorphism. In particular, it is defined for any group ring $ \mathbf Z [ \pi _ {1} ( X)] $,
7 KB (997 words) - 08:28, 6 June 2020
...y of polycyclic groups. If $k$ is an algebraic extension of a finite field and $G$ is a finite extension of a polycyclic group, then any simple $kG$-modul
...[2]</TD> <TD valign="top"> B.A.F. Wehrfritz, "Three lectures on polycyclic groups" , Queen Mary College London (1973)</TD></TR></table>
4 KB (588 words) - 19:59, 11 April 2014
...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