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**On \(p\)-injectivity, YJ-injectivity and quasi-Frobeniusean rings.**
*(English)*
Zbl 1068.16004

The main results of the paper are the following. A ring \(A\) is von Neumann regular if and only if \(A\) is a semiprime ring whose finitely generated one-sided ideals are annihilators of an element of \(A\) and this is equivalent to the condition that \(A\) is a semiprime ring such that every finitely generated left ideal is the left annihilator of an element of \(A\) and every principal right ideal of \(A\) is the right annihilator of an element of \(A\) (Theorem 1). Further, \(A\) is a left Artinian ring if and only if it is a left Noetherian ring such that for any prime factor-ring \(B\) of \(A\), every essential right ideal of \(B\) is an idempotent two-sided ideal of \(B\) (Theorem 5).

Recall, that a left \(A\)-module is \(p\)-injective if it is injective with respect to the embeddings of principal left ideals of \(A\) into \(A\). The following conditions are equivalent for a ring \(A\): (1) \(A\) is quasi-Frobenius; (2) \(A\) is left pseudo-Frobenius and projective \(p\)-injective right \(A\)-modules are injective; (3) there exists a \(p\)-injective left generator of \(A\)-Mod and projective \(p\)-injective left modules are injective (Theorem 9).

Recall that a left \(A\)-module \(M\) is YJ-injective if, for any \(0\neq a\in A\), there exists a positive integer \(n\) with \(a^n\neq 0\) such that every left \(A\)-homomorphism of \(Aa^n\) into \(M\) extends to \(A\). A ring \(A\) is quasi-Frobenius if and only if \(A\) is YJ-injective ring with maximum condition on annihilators and the socle of \(A\) is finitely generated (Theorem 11).

Recall, that a left \(A\)-module is \(p\)-injective if it is injective with respect to the embeddings of principal left ideals of \(A\) into \(A\). The following conditions are equivalent for a ring \(A\): (1) \(A\) is quasi-Frobenius; (2) \(A\) is left pseudo-Frobenius and projective \(p\)-injective right \(A\)-modules are injective; (3) there exists a \(p\)-injective left generator of \(A\)-Mod and projective \(p\)-injective left modules are injective (Theorem 9).

Recall that a left \(A\)-module \(M\) is YJ-injective if, for any \(0\neq a\in A\), there exists a positive integer \(n\) with \(a^n\neq 0\) such that every left \(A\)-homomorphism of \(Aa^n\) into \(M\) extends to \(A\). A ring \(A\) is quasi-Frobenius if and only if \(A\) is YJ-injective ring with maximum condition on annihilators and the socle of \(A\) is finitely generated (Theorem 11).

Reviewer: Ladislav Bican (Praha)

### MSC:

16D50 | Injective modules, self-injective associative rings |

16L60 | Quasi-Frobenius rings |

16E50 | von Neumann regular rings and generalizations (associative algebraic aspects) |

16N60 | Prime and semiprime associative rings |