...nd multiplication of them by elements of $K$ are defined in the usual way, and the commutator $[x,y]$ of two elements $x,y \in \mathfrak{g}$ is given by t
...rally identified with the set of all square matrices of order $n$ over $K$ and is denoted by $\mathfrak{gl}(n,K)$. Any linear Lie algebra is a subalgebra
3 KB (554 words) - 19:21, 24 February 2017
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...role in ring theory. Every ring with identity has maximal left (also right and two-sided) ideals. The quotient module $ M = R / I $
7 KB (1,160 words) - 08:00, 6 June 2020
...— associativity; this is the explanation of the term "semi-group" . Semi-groups are called ''monoids'' if they have, in addition, an identity element.
...ps of partial transformations), and the algebraic theory of automata (semi-groups of automata).
17 KB (2,435 words) - 09:18, 2 April 2023
its semi-simple and nilpotent components $s$ and $n$ should lie in ${\mathfrak g}$
case of a field $k$ of characteristic 0, a necessary and sufficient
3 KB (458 words) - 04:43, 4 January 2022
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...A}$ an [[Abelian category|Abelian category]] with exact infinite products, and $M : \mathcal{C} \rightarrow \mathcal{A}$ a covariant [[Functor|functor]].
9 KB (1,283 words) - 20:55, 8 February 2024
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is defined) and equivariant mappings is a sequence of contravariant functors $ H _ {G} ^
5 KB (772 words) - 11:12, 20 January 2021
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and is an acyclic mapping. This means that $ q _ {N} $
7 KB (1,010 words) - 16:47, 17 March 2023
...functors on various categories of algebraic objects (modules over a given ring, sheaves, etc.).
...th the difference that the transition from geometry to algebra in homology theory is irreversible).
12 KB (1,885 words) - 23:48, 23 April 2017
...these theorems usually become invalid, and the Grothendieck group $K_0(R)$ and the Whitehead group $K_1(R)$ are, in a certain sense, a measure of their de
...jects may be studied with the aid of the homotopy theory of vector bundles and of topological
14 KB (2,405 words) - 22:14, 10 January 2015
...def\phi{\varphi}\phi:X\times X \to R$ that is linear in the first argument and satisfies the condition
Here $R$ is a [[unital ring|ring with a unit element]] and equipped with an involutory [[anti-automorphism]] $J$. In particular, $\phi
5 KB (831 words) - 17:13, 9 October 2016
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into the category of groups Gr. A homomorphism of a group object $ X $
6 KB (864 words) - 19:42, 5 June 2020
...of semi-linear transformations of $E$ preserving $f$ up to a scalar factor and an automorphism of $K$).
...be characterized as groups of those transformations of projective spaces (and also of certain varieties related to Grassmannians, see
12 KB (1,991 words) - 18:05, 29 November 2014
...ariski topology), is not a topological group with respect to this topology and therefore is not a compact group.
The following groups are two important classes of compact groups.
8 KB (1,117 words) - 20:03, 27 February 2021
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and such that all its local homology groups (cf. [[Local homology|Local homology]]) $ H _ {p} ^ {x} $
9 KB (1,445 words) - 22:10, 5 June 2020
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be an associative ring with unit element. A sequence of elements $ ( a _ {1} \dots a _ {n} ) $
7 KB (1,046 words) - 08:22, 6 June 2020
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of an algebra (group, ring, etc.) $ A $
2 KB (242 words) - 16:40, 31 March 2020
.... also [[Pro-p group|Pro-$p$-group]]; [[Cohomology of groups|Cohomology of groups]]). The original proof appeared in [[#References|[a7]]], a proof in the con
...extbf{Z}/p\to\textbf{Z}/p^2\to\textbf{Z}/p\to 0$ (see [[#References|[a9]]] and [[Cohomology operation|Cohomology operation]]). Note that for $p=2$ this is
6 KB (868 words) - 22:16, 5 February 2021
...bilinear form $\Phi$ on a left $K$-module $E$, where $K$ is a commutative ring (cf.
[[Classical group|Classical group]]). In the case when $E=K^{2m}$ and the matrix of $\Phi$ with respect to the canonical basis $\{e_i\}$ of $E$ h
6 KB (1,078 words) - 14:22, 3 November 2013
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and $ F _ {2} $
4 KB (558 words) - 16:18, 6 June 2020
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be a [[Scheme|scheme]] and let $ p $
6 KB (885 words) - 08:04, 6 June 2020