Modules which are invariant under idempotents of their envelopes
Volume 143 / 2016
Colloquium Mathematicum 143 (2016), 237-250
MSC: Primary 16D50; Secondary 16D80, 16W20.
DOI: 10.4064/cm6496-1-2016
Published online: 14 January 2016
Abstract
We study the class of modules which are invariant under idempotents of their envelopes. We say that a module $M$ is $\mathcal{X}$-idempotent-invariant if there exists an $\mathcal{X}$-envelope $u : M \rightarrow X$ such that for any idempotent $g\in \operatorname{End}(X)$ there exists an endomorphism $f : M \rightarrow M$ such that $uf = gu$. The properties of this class of modules are discussed. We prove that $M$ is $\mathcal{X}$-idempotent-invariant if and only if for every decomposition $X=\bigoplus_{i\in I}X_i$, we have $M=\bigoplus_{i\in I} (u^{-1}(X_i)\cap M)$. Moreover, some generalizations of $\mathcal{X}$-idempotent-invariant modules are considered.
