Left perfect rings that are right perfect and a characterization of Steinitz rings

Abstract

A proof is given to show all flat left modules of a ring are free if and only if the ring is a local ring with a left Tnilpotent maximal ideal. We characterize left perfect rings whose radical R has the property that IRn={O} for some positive integer n if 1 is a finitely generated right ideal contained in R. We cite an example of a left perfect ring which does not have this property. It is shown that if the set of irreducible elements of a left perfect ring is right T-nilpotent then the ring is right perfect. Introduction. If R is the radical of a left perfect ring, an element of R is called irreducible if it cannot be expressed as a product of two elements of R. If R is the radical of a ring that is left and right perfect, we show each element of R can be expressed as a product of irreducible elements. It follows that a perfect ring having a finite set of irreducible elements is right and left artinian and has a finite radical; furthermore if the ring is a local ring with a nonzero radical, the ring is finite. Throughout this paper we will assume that a ring has an identity element and that a module is unitary. DEFINITION. Let F be a free left A-module having {uj}1=1 as a basis; let {aj}n1 be a sequence in A and let G denote the submodule of F generated by {Uj-ajuj+1} 1. The pair Fand G will be denoted [F, {aj}, G]. LEMMA 1. FIG is a flat left A-module. PROOF. It is sufficient to show that if I is a right ideal, IFnG=IG. If mI Secondary 16A46.