Finitely Presented Modules Over Semiperfect Rings

Abstract

Results of Bjork and Sabbagh are extended and generalized to provide a Krull-Schmidt theory over a general class of semiperfect rings which includes left perfect rings, right perfect rings, and semiperfect Pi-rings whose Jacobson radicals are nil.The object of this paper is to lay elementary foundations to the study of f.g.(i.e.finitely generated) modules over rings which are almost Artinian, with the main goal being a theory following the lines of the Azumaya-Krull-Remak-Schmidt-Wedderburn theorem (commonly called Krull-Schmidt); in other words we wish to show that a given f.g.module is a finite direct sum of indecomposable submodules whose endomorphism rings are local.Previous efforts in this direction include [2,3,4,6], and in particular the results here extend some results of [2,4,6].The focus here will be on a "Fitting's lemma" approach applied to semiperfect rings, cf.Theorem 8.We recall the definition from [1], which will be used as a standard reference.R is semiperfect if its Jacobson radical J is idempotent-lifting and R/J is semisimple Artinian; equivalently every f.g.module M has a projective cover (an epic map 7t: P -* M, where P is projective and kerrr is a small submodule of P).Projective covers are unique up to isomorphism by [1, Lemma 17.17].In what follows, module means "left module".PROPOSITION 1.If R is semiperfect, then every f.g.module M is a finite direct sum of indecomposable submodules.PROOF.Let it: P -> M be a projective cover.Then P has an indecomposable decomposition of some length (cf.[1, Theorem 27.12]) and we show by induction on t that M also has an indecomposable decomposition of length < t.Indeed this is tautological if M is indecomposable, so assume M = M\ © M2-By [1, Lemma 17.17] there are projective covers ixt : Pi -* Mi, where P¿ are direct summands of P, and in fact Pi © P2 « P by [1, Exercise 15.1], so we can proceed inductively on Mi and M2.Q.E.D. REMARK 2. By [1, Theorem 27.6] an f.g..ß-moduleM is a direct sum of (indecomposable) modules having local endomorphism iff End^M is semiperfect, so we ask: For which modules M is End#M semiperfect?(This is why it is natural to study semiperfect rings R.) In [4, Example 2.1] Bjork found an example of a cyclic module M = R/L over a semiprimary ring R such that E = End^M is not