CONTINUOUS RINGS WITH CHAIN-CONDITIONS

The quasi-Frobenius rings are characterized as the left continuous rin gs satisfying either (A1) or (A2) and either (S1) or (S2), where these conditions are defined as follows: (A1): ACC on left annihalitors; (A 2): R/Soc((R)R) is left Goldie; (S1): S = r(l(S)) for every minimal ri ght ideal S; and (S2): Every minimal right ideal is essential in a sum mand of R(R). These characterizations extend several results in the li terature. In addition, it is shown that, in these rings, Soc(R(R)) = S oc((R)R), Soc(eR) is simple for every primitive indempotent e of R, an d there exists a complete set of distinct representatives {Rt1,...,Rt( n)} of the isomorphism classes of the simple left R-modules such that {t1R,...,t(n)R} is a complete set of distinct representatives of the i somorphism classes of the simple right R-modules.