A characterization of perfect rings

Abstract

J. P. Jans has shown that if a ring R is right perfect, then a certain torsion in the category Mod R of left ϋί-modules is closed under taking direct products. Extending his method, J. S. Alin and E. P. Armendariz showed later that this is true for every (hereditary) torsion in Mod R. Here, we offer a very simple proof of this result. However, the main purpose of this paper is to present a characterization of perfect rings along these lines: A ring R is right perfect if and only if every (hereditary) torsion in Mod R is fundamental (i.e., derived from prime torsions) and closed under taking direct products; in fact, then there is a finite number of torsions, namely 2 for a natural number n. Finally, examples of rings illustrating that the above characterization cannot be strengthened are provided. Thus, an example of a ring R± is given which is not perfect, although there are only fundamental torsions in Mod Ri, and only 4 = 2 of these. Furthermore, an example of a ring R2* is given which is not perfect and which, at the same time, has the property that there is only a finite number (namely, 3) of (hereditary) torsions in Mod i?2* all of which are closed under taking direct products. Moreover, the ideals of R2* form a chain (under inclusion) and Rad R2* is a nil idempotent ideal; all the other proper ideals are nilpotent and R2* can be chosen to have a (unique) minimal ideal.