Modules which are invariant under idempotents of their envelopes

Le Van Thuyet ¹   ; Phan Dan ²   ; Truong Cong Quynh ³

¹ Department of Mathematics Hue University 3 Le Loi Hue, Vietnam
² Banking University of Ho Chi Minh City 39 Ham Nghi Ho Chi Minh City, Vietnam
³ Department of Mathematics Danang University 459 Ton Duc Thang DaNang, Vietnam

Colloquium Mathematicum, Tome 143 (2016) no. 2, pp. 237-250

Résumé

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.

Détail

DOI : 10.4064/cm6496-1-2016

Keywords: study class modules which invariant under idempotents their envelopes say module mathcal idempotent invariant there exists mathcal envelope rightarrow idempotent operatorname end there exists endomorphism rightarrow properties class modules discussed prove mathcal idempotent invariant only every decomposition bigoplus have bigoplus cap moreover generalizations mathcal idempotent invariant modules considered