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PI-injective modules. (Chinese. English summary) Zbl 1224.16004

Summary: The notion of PI-injective modules is introduced, which is a natural generalization of cotorsion modules. By discussing the properties of PI-injective modules, weakly perfect ring is defined and some new characterizations of von Neumann regular rings and Noetherian rings are given. The following results are proved: (1) Let \(R\) be a ring. If \(R\) is right Noetherian, then each right \(R\)-module is PI-injective; (2) Let \(R\) be a Noetherian ring, then \(R\) is perfect if and only if each PI-injective module is cotorsion.

MSC:

16D50 Injective modules, self-injective associative rings
16P40 Noetherian rings and modules (associative rings and algebras)
16L30 Noncommutative local and semilocal rings, perfect rings
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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