Homological Properties of PS-Rings

Let R be a ring, S = Soc(R_R). R is called a PS-ring if S_R is projective. In this paper, we will give a characterization of the PS-rings by showing the following results:Theorem 1.4 The following statements are equivalent for a ring R:(1) R is a. PS-ring;(2) S = S;(3) For every homogenous component H_i, of S_R, R/H_i is a flat left R- module;(4) R/S is a flat left R-module.Theorem 2.4 Let R be a PS-ring, R = R/S,and M a right .R-module. Then rid_R(M=rid_RM)).Corollary 2.8 Let R be a primitive ring. Then WD(R) = WD(R/S).