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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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References:
[1] Andersony F.W., Rings and Categories of Modules (1992)
[2] DOI: 10.1007/BF02767358 · Zbl 0667.13005
[3] DOI: 10.1006/jabr.1993.1195 · Zbl 0802.16003
[4] DOI: 10.1016/0021-8693(80)90113-1 · Zbl 0477.16013
[5] DOI: 10.1090/S0002-9939-1994-1209416-4
[6] DOI: 10.1016/0021-8693(90)90252-J · Zbl 0701.16023
[7] DOI: 10.1090/S0002-9947-1960-0120260-3
[8] DOI: 10.1007/BF02567857 · Zbl 0802.16023
[9] DOI: 10.1007/BF02760849 · Zbl 0464.16019
[10] Enochs E.E., Private communication
[11] DOI: 10.1016/0021-8693(67)90035-X · Zbl 0173.03203
[12] Faith C., Lecture Notes in Math 951 pp 21– (1982)
[13] DOI: 10.1080/00927878908823712 · Zbl 0667.16021
[14] DOI: 10.1007/BF01223887 · Zbl 0356.16006
[15] Kasch F., Modules and Rings (1982) · Zbl 0523.16001
[16] Rotman J.J., An Introduction to Homological Algebra (1979) · Zbl 0441.18018
[17] DOI: 10.1070/SM1978v034n02ABEH001155 · Zbl 0402.16022
[18] Stenstrom B., Rings of Quotients (1975)
[19] DOI: 10.1112/jlms/s2-2.2.323 · Zbl 0194.06602
[20] Wisbauer R., Algebra, Logic and Applications 3 3 (1991)
[21] DOI: 10.1090/S0002-9939-1995-1242111-5
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