Semiperfect min-CS rings

Abstract

We show that if R is a ring such that each minimal left ideal is essential in a (direct) summand of _RR, then the dual of each simple right R-module is simple if and only if R is semiperfect with Soc(_RR)=Soc(R_R) and Soc(Re) is simple and essential for every local idempotent e of R. We also show that R is left CS and right Kasch if and only if R is a semiperfect left continuous ring with Soc(R_R)⊆^e_RR. As a particular case of both results we obtain that R is a ring such that every (essential) closure of a minimal left ideal is summand (R is then said to be left strongly min-CS) and the dual of each simple right R-module is simple if and only if R is a semiperfect left continuous ring with Soc(_RR)=Soc(R_R)⊆^e_RR. Moreover, in this case R is also left Kasch, Soc(eR)≠0 for every local idempotent e of R, and R admits a (Nakayama) permutation of a basic set of primitive idempotents. As a consequence of this result we characterise left PF rings in terms of simple modules over the 2×2 matrix ring by showing that R is left PF if and only if M₂(R) is a left strongly min-CS ring such that the dual of every simple right module is simple.