Semiperfect min-CS rings

José L. Gómez Pardo, Mohamed F. Yousif
1999 Glasgow Mathematical Journal  
We show that if R is a ring such that each minimal left ideal is essential in a (direct) summand of R R, then the dual of each simple right R-module is simple if and only if R is semiperfect with o R R oR R and oRe 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 oR R e R R. As a particular case of both results we obtain that R is a ring such that every (essential) closure of a
more » ... ial) 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 o R R oR R e R R. Moreover, in this case R is also left Kasch, oeR T 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 2Â2 matrix ring by showing that R is left PF if and only if M 2 R is a left strongly min-CS ring such that the dual of every simple right module is simple. 1. Introduction. An important source of semiperfect rings is given by the theorem of B. Osofsky [14] which asserts that a left injective cogenerator ring (also called a left PF ring) is semiperfect and has ®nitely generated essential left socle. Conversely, if R is left self-injective, semiperfect, and has essential left socle, then R is left PF [15, 48.12]. It is obvious that R is a left PF ring if and only if it is left selfinjective and left Kasch, where the latter condition just means that every simple left R-module is isomorphic to a (minimal) left ideal. From Osofsky's theorem it also follows that a left PF ring is right Kasch and so it is natural to ask whether a left self-injective right Kasch ring is left PF. This problem is still open but in order to obtain a positive solution it would be enough to prove that R has essential socle, because it has already been shown in [6] that these rings are semiperfect. This result was extended in [18] , where it was shown that if R is left CS and the dual of every simple right R-module is simple, then R is semiperfect with o R R oR R e R R. In this paper we look for the weakest conditions of this type that imply the ring is semiperfect. Instead of left CS rings we consider the much larger class of left min-CS rings (cf. [13]), i.e., rings R such that every minimal left ideal is essential in a direct summand and we show that this weak injectivity property is useful to obtain semiperfect rings. Indeed, we prove in Theorem 2.1 that if R is left min-CS, then the dual of every simple right R-module is simple if and only if R is semiperfect with o R R oR R and oRe is simple and essential for every local idempotent e of R. Thus we establish the following pattern: we work with an injectivity condition
doi:10.1017/s0017089599970878 fatcat:iujy45brqnbvjaq4vxabn2zbvq