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Semiperfect modules relative to a torsion theory. (English) Zbl 0608.16024

A projective module P is semiperfect when every factor module of P has a projective cover. E. A. Mares has shown [Math. Z. 82, 347-360 (1963; Zbl 0131.27401)] that H. Bass’ characterization of semiperfect rings [given in Trans. Am. Math. Soc. 95, 466-488 (1960; Zbl 0094.022)] carries over to semiperfect modules. Specifically, Mares has shown that a module P is semiperfect if and only if Rad(P) is small in P, P/Rad(P) is semisimple and every direct decomposition of P/Rad(P) lifts to a direct decomposition of P.
This paper is devoted to studying semiperfect modules relative to a hereditary torsion theory in the category R-Mod of unitary left R-modules with corresponding left Gabriel topology \({\mathcal F}\). A module M is strongly \({\mathcal F}\)-projective if every \({\mathcal F}\)-dense submodule is \({\mathcal F}\)-projective where \({\mathcal F}\)-projectivity is defined in the sense of O. Goldman [J. Algebra 13, 10-47 (1969; Zbl 0201.040)]. The authors define strongly \({\mathcal F}\)-projective covers and call a module M \({\mathcal F}\)-semiperfect when every factor module of M has a strongly \({\mathcal F}\)-projective cover. This is equivalent to \(M_{{\mathcal F}}\), the module of quotients of M, being semiperfect in the full subcategory (R,\({\mathcal F})\)-Mod of \({\mathcal F}\)-torsion free and \({\mathcal F}\)- injective modules. One result the authors obtain is the following characterization of \({\mathcal F}\)-semiperfect modules which reduces to Mares’ result when the torsion theory in R-Mod is the torsion theory in which every module is torsion free.
Theorem. Let P be a strongly \({\mathcal F}\)-projective module and assume that the following hold: (a) \(P/Rad_{{\mathcal F}}(P)\) is \({\mathcal F}\)-semisimple. (b) \(Rad_{{\mathcal F}}(P)\) is \({\mathcal F}\)-superfluous in P. (c) Every \({\mathcal F}\)-decomposition (or finite \({\mathcal F}\)-decomposition) \(\{M_ i\}\) of \(P/Rad_{{\mathcal F}}(P)\), with each \(M_ i\) saturated in \(P/Rad_{{\mathcal F}}(P)\) lifts with respect to the canonical projection \(\epsilon\) : \(P\to P/Rad_{{\mathcal F}}(P)\). Then P is an \({\mathcal F}\)- semiperfect module. If, moreover, (R,\({\mathcal F})\)-Mod is locally finitely generated, then the converse holds.
The authors also study \({\mathcal F}\)-semiperfect rings \((=rings\) for which \({}_ RR\) is an \({\mathcal F}\)-semiperfect module) and obtain the following characterization of \({\mathcal F}\)-semiperfect rings.
Theorem. Let \({\mathcal F}\) be a Gabriel topology which has a basis of finitely generated left ideals. Then the following are equivalent: (a) R is \({\mathcal F}\)-semiperfect. (b) Every finitely generated object of (R,\({\mathcal F})\)-Mod has a projective cover in (R,\({\mathcal F})\)-Mod. (c) R has an \({\mathcal F}\)-projective cover and every \({\mathcal F}\)-cocritical quotient of R has an \({\mathcal F}\)-projective cover. (d) There exists an \({\mathcal F}\)-projective cover \(\epsilon\) : \(P\to R\) of R such that P has an \({\mathcal F}\)-decomposition by \({\mathcal F}\)-local modules.
Reviewer: P.E.Bland

MSC:

16L30 Noncommutative local and semilocal rings, perfect rings
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
16D40 Free, projective, and flat modules and ideals in associative algebras
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