On injectivity and \(p\)-injectivity. III (Q2777991)

[For part II cf. Soochow J. Math. 21, No. 4, 401-412 (1995; Zbl 0840.16007).]NEWLINENEWLINENEWLINEThe author studies \(p\)-injective modules and characterizes quasi-Frobeniusean rings in terms of \(p\)-injective modules. Main results are: (i) For a commutative ring \(A\), every factor ring is quasi-Frobeniusean if and only if every factor ring of \(A\) is \(p\)-injective with maximum condition on annihilators. (ii) Every factor ring of a ring \(A\) is left self injective regular with nonzero socle if and only if every factor ring of \(A\) is semiprime with an injective maximal left ideal. (iii) If \(A\) is a right Noetherian ring whose factor rings are left \(p\)-injective then \(A\) is right Artinian.