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On endomorphism rings of quasiprojective modules. (English) Zbl 0562.16012

Let M be a \(\Sigma\)-quasiprojective R-module, i.e., a module such that any direct sum of copies of M is quasiprojective. The endomorphism ring S of M is studied by means of the category equivalence that exists between the quotient category of S-mod by the Gabriel topology of all the left ideals I of S such that \(MI=M\) and a suitable quotient category of the category \(\sigma\) [M] of all the submodules of R-modules generated by M. This technique is useful to obtain necessary and sufficient conditions on M for S to have a given property and it is applied here to a variety of situations. For instance, conditions on M are given (sometimes under more restrictive hypotheses, such as M being finitely generated) for S to be a left V-ring or to have other properties related to several chain conditions. It is also shown that if M is \(\Sigma\)-quasiprojective, then the dual Goldie dimensions [K. Varadarajan, Commun. Algebra 7, 565- 610 (1979; Zbl 0487.16019)] of S and M are the same.

MSC:

16D40 Free, projective, and flat modules and ideals in associative algebras
16S50 Endomorphism rings; matrix rings
16D90 Module categories in associative algebras
16P60 Chain conditions on annihilators and summands: Goldie-type conditions

Citations:

Zbl 0487.16019
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Full Text: DOI

References:

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