[For part I cf. J. Math. Kyoto Univ. 27, 439-452 (1987; Zbl 0655.16012).] A left \(A\)-module \(M\) is called \(p\)-injective if for every principal left ideal \(P\) of \(A\), every \(A\)-homomorphism from \(P\) to \(M\) can be extended to \(A\) to \(M\). In this paper the author shows: (1) If \(A\) is a semi-prime principal left ideal ring such that every nonzero complement left ideal contains a nonzero two sided ideal, then divisible left \(A\)-modules are injective. (2) If \(A\) contains an injective maximal left ideal and any minimal non-nilpotent left ideal is a two-sided ideal of \(A\), then \(A\) is left-injective. (3) If every complement left ideal of \(A\) is a two-sided ideal and every simple left \(A\)-module is either \(p\)-injective or projective, then \(A\) is a fully left idempotent ring whose complement left ideals are left annihilators. (4) \(A\) is left Artinian if, and only if, \(A\) is left Noetherian such that each prime factor ring \(B\) satisfies any one of the following conditions: (a) \(B\) contains a finitely generated \(p\)-injective maximal left ideal; (b) every essential left ideal of \(B\) is an idempotent two-sided ideal.