The rings in the title are studied and related to right principally injective rings. Many properties of these rings (called left pseudo-morphic by Yang) are derived, and conditions are given that an endomorphism ring is left pseudo-morphic. Some particular results: (1) Commutative pseudo-morphic rings are morphic; (2) Semiprime left pseudo-morphic rings are semisimple; and (3) A left and right pseudo-morphic ring satisfying (equivalent) mild finiteness conditions is a morphic, quasi-Frobenius ring in which every onesided ideal is principal. Call a left ideal L a left principal annihilator if L = l(a) = {r ∈ R | ra = 0} for some a ∈ R. It is shown that if R is left pseudo-morphic, left mininjective ring with the ACC on left principal annihilators then R is a quasi-Frobenius ring in which every right ideal is principal and every left ideal is a left principal annihilator.