Modules which are invariant under idempotents of their envelopes (Q2804265)

\textit{P. A. Guil Asensio} et al. initiated the study of modules that are invariant under automorphisms of their general envelopes and covers in [Isr. J. Math. 206, 457--482 (2015; Zbl 1337.16002)]. In the particular case of injective envelopes, it was studied earlier by \textit{S. E. Dickson} and \textit{K. R. Fuller} [Pac. J. Math. 31, 655--658 (1969; Zbl 0185.09301)]. The article under review studies modules invariant under idempotent endomorphisms of their envelopes. Let \(\mathcal X\) be an enveloping class. A module \(M\) with an \(\mathcal X\)-envelope \(u:M\rightarrow X\) is called \(\mathcal X\)-idempotent invariant if for any idempotent \(g\in \mathrm{End}(X)\) there exists \(f\in \mathrm{End}(M)\) such that \(u \circ f=g \circ u\). The authors obtain some basic properties of this class of modules. In case \(\mathcal X\) is the class of injective modules, \(\mathcal X\)-idempotent invariant modules are precisely the quasi-continuous (or \(\pi\)-injective) modules.