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We study five different types of the homology of a Lie algebra over a commutative ring which are naturally isomorphic over fields. We show that they are not isomorphic over commutative rings, even over $mathbb Z,$ and study connections between them. In particular, we show that they are naturally isomorphic in the case of a Lie algebra which is flat as a module. As an auxiliary result we prove that the Koszul complex of a module $M$ over a principal ideal domain that connects the exterior and the symmetric powers $0to Lambda^n Mto M otimes Lambda^{n-1} M to dots to S^{n-1}M otimes M to S^nMto 0 $ is purely acyclic.
We discuss a version of the Chevalley--Eilenberg cohomology in characteristic $2$, where the alternating cochains are replaced by symmetric ones.
We describe new classes of noetherian local rings $R$ whose finitely generated modules $M$ have the property that $Tor_i^R(M,M)=0$ for $igg 0$ implies that $M$ has finite projective dimension, or $Ext^i_R(M,M)=0$ for $igg 0$ implies that $M$ has fini
The aim of this paper is to introduce and study Lie algebras and Lie groups over noncommutative rings. For any Lie algebra $gg$ sitting inside an associative algebra $A$ and any associative algebra $FF$ we introduce and study the algebra $(gg,A)(FF)$
For a commutative ring R with many units, we describe the kernel of H_3(inc): H_3(GL_2(R), Z) --> H_3(GL_3(R), Z). Moreover we show that the elements of this kernel are of order at most two. As an application we study the indecomposable part of K_3(R).
Let $M$ be an $R$-module and $c$ the function from $M$ to the ideals of $R$ defined by $c(x) = cap lbrace I colon I text{is an ideal of} R text{and} x in IM rbrace $. $M$ is said to be a content $R$-module if $x in c(x)M $, for all $x in M$. $B$ is c