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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.

##### MSC:
 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)
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