On a Class of Perfect Rings

In [3], the perfect rings of Bass [1] were characterized in terms of torsions in the following way: A ring R is right perfect if and only if every (hereditary) torsion in the category Mod R of all left R-modules is fundamental (i.e. generated by some minimal torsions) and closed under taking direct products; as a consequence, the number of all torsions in Mod R is finite and equal to 2 n for a natural n. Here, we present a simple description of those rings R which allow only two (trivial) torsions, viz. 0 and Mod R (and thus, are right perfect by [3]). Finite direct sums of these rings represent a natural generalization of completely reducible (i.e. artinian semisimple) rings (cf. Theorem 2) and we shall call them for that matter π-reducible rings. They can also be characterized in terms of their idempotent two-sided ideals.