Let $R$ be a commutative ring, $M$ an $R$-module and $G$ a group of $R$-automorphisms of $M$, usually with some sort of rank restriction on $G$. We study the transfer of hypotheses between $M/C_M(G)$ and $[M,G]$ such as Noetherian or having finite composition length. In this we extend recent work of Dixon, Kurdachenko and Otal and of Kurdachenko, Subbotin and Chupordia. For example, suppose $[M,G]$ is $R$-Noetherian. If $G$ has finite rank, then $M/C_M(G)$ also is $R$-Noetherian. Further, if $[M,G]$ is $R$-Noetherian and if only certain abelian sections of $G$ have finite rank, then $G$ has finite rank and is soluble-by-finite. If $M/C_M(G)$ is $R$-Noetherian and $G$ has finite rank, then $[M,G]$ need not be $R$-Noetherian., Bertram A. F. Wehrfritz., and Obsahuje bibliografické odkazy
Let $X$ be a Banach space. We give characterizations of when ${\cal F}(Y,X)$ is a $u$-ideal in ${\cal W}(Y,X)$ for every Banach space $Y$ in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when ${\cal F}(X,Y)$ is a $u$-ideal in ${\cal W}(X,Y)$ for every Banach space $Y$, when ${\cal F}(Y,X)$ is a $u$-ideal in ${\cal W}(Y,X^{**})$ for every Banach space $Y$, and when ${\cal F}(Y,X)$ is a $u$-ideal in ${\cal K}(Y,X^{**})$ for every Banach space $Y$.