Σ-semi-compact rings and modules

Abstract

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 (respectively, descending) chain condition on the left (respectively, 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 Min R compact. We also prove that every (ℵ0, 1)-flat left R-module is singly projective if R is left Σ-semi-compact, and the converse holds if Rℕ is an (ℵ0, 1)-flat left R-module.