On injective modules and annihilators
Let A be an associative ring with identity. The ring A is called V-ring iff every simple left A-module is injective. A is called a left PA-ring if for any principal left ideal P of A, there exists a positive integer n such that \(P^ n\) is a left annihilator ideal. A left A-module M is called CE-injective if for any left submodule N, containing a non-zero complement left submodule of M, every left A- homomorphism of N into M extends to an endomorphism of \({}_ AM\). A is called left CE-injective if \({}_ AA\) is CE-injective. A is called ELT if every essential left ideal is twosided. The author considers some generalizations of injectivity. He proves for an ELT ring A that the following conditions are equivalent: (I) A is a left and right selfinjective regular, left and right V-ring of bounded index; (2) A is a semisimple left or right PA, left CE-injective ring; (3) A is a fully left idempotent left CE-injective ring. The author investigates new additional characteristic properties of semisimple Artinian rings [cf. the author, Math. Jap. 19, 173-176 (1974; Zbl 0263.16019)]. Among them the following conditions are equivalent: (I) A is semisimple Artinian; (2) A is ELT such that for any non-zero left ideal I, there exists a positive integer n such that \(I^ n\) is a non- zero projective left annihilator; (3) For any non-zero left ideal I of A, there exists a positive integer n such that \(I^ n\) is a non-zero direct summand of \({}_ AA\); (4) A is either left or right p-injective such that the direct sum of any two CE-injective left A-modules is CE- injective. Considering a ring A with linearly ordered left ideals (LO-ring), the author proves that A is a local left principal ideal quasi-Frobenius ring iff A is a PA, LO-ring with maximum condition on annihilators.
scientific article; zbMATH DE number 3922896
scientific article