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Rings whose class of projective modules is socle fine. (English) Zbl 1066.16004

Summary: A class \(\mathcal C\) of modules over a unitary ring is said to be socle fine if whenever \(M,N\in{\mathcal C}\) with \(\text{Soc}(M)\cong\text{Soc}(N)\) then \(M\cong N\). In this work we characterize certain types of rings by requiring a suitable class of its modules to be socle fine. Then we study socle fine classes of quasi-injective, quasi-projective and quasicontinuous modules which we apply to find socle fine classes in special types of Noetherian rings. We also initiate the study of those rings whose class of projective modules is socle fine.

MSC:

16D80 Other classes of modules and ideals in associative algebras
16D40 Free, projective, and flat modules and ideals in associative algebras
16D50 Injective modules, self-injective associative rings
13C13 Other special types of modules and ideals in commutative rings
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