$Σ$-semi-compact rings and modules

In this paper several characterizations of semi-compact modules are given. Among other results, we study rings whose semi-compact modules are injective. We introduce the property $Σ$-semi-compact for modules and we characterize the modules satisfying this property. In particular, we show that a ring $R$ is left $Σ$-semi-compact if and only if $R$ satisfies the ascending (resp. descending) chain condition on the left (resp. right) annulets. Moreover, we prove that every flat left $R$-module is semi-compact if and only if $R$ is left $Σ$-semi-compact. We also show that a ring $R$ is left Noetherian if and only if every pure projective left $R$-module is semi-compact. Finally, we consider rings whose flat modules are finitely (singly) projective. For any commutative arithmetical ring $R$ with quotient ring $Q$, we prove that every flat $R$-module is semi-compact if and only if every flat $R$-module is finitely (singly) projective if and only if $Q$ is pure semisimple. A similar result is obtained for reduced commutative rings $R$ with the space $\mathrm{Min}\ R$ compact. We also prove that every $(\aleph_{0},1)$-flat left $R$-module is singly projective if $R$ is left $Σ$-semi-compact, and the converse holds if $R^{\mathbb{N}}$ is an $(\aleph_{0},1)$-flat left $R$-module.

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