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Rings characterized by (pre)envelopes and (pre)covers of their modules. (English) Zbl 0908.16003
The authors generalize the work of E. E. Enochs [Isr. J. Math. 39, 189-209 (1981; Zbl 0464.16019)], and others, on (pre)envelopes and (pre)covers of modules. The basic notion introduced is the following: A class of \(R\)-modules \(\mathcal F\) is locally initially small if for every \(R\)-module \(M\), there exists a set \({\mathcal F}_M\subseteq{\mathcal F}\) such that every homomorphism \(M\to F\), where \(F\in{\mathcal F}\), factors through a direct product of modules in \({\mathcal F}_M\). The dual notion is also discussed. Among the many results obtained, the following is typical: every \(R\)-module has an \(\mathcal F\)-preenvelope if and only if \(\mathcal F\) is locally initially small and every direct product of modules in \(\mathcal F\) is a direct summand of a module in \(\mathcal F\).
When applied to classes of flat, projective and free \(R\)-modules, the theory yields various characterizations of rings of which the following is a sample: A ring \(R\) is semiprimary, left hereditary and right coherent if and only if every left \(R\)-module has an epimorphic projective envelope.

16D40 Free, projective, and flat modules and ideals in associative algebras
16D50 Injective modules, self-injective associative rings
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
Full Text: DOI
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