J. P. Jans has shown that if a
ring R is right perfect, then a certain torsion in the category Mod R of left
R-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 2n 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 R1 is given which is not perfect,
although there are only fundamental torsions in Mod R1, and only 4 = 22 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 R2∗ 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.
