FACTOR RINGS OF PSEUDO-FROBENIUS RINGS

Abstract

If R is right pseudo-Frobenius (= PF), and A is an ideal, when is R/A right PF? Our main result, Theorem 3.7, states that this happens iff the ideal A′ of the basic ring B of R corresponding to A has left annihilator F in B generated by a single element on both sides. Moreover, in this case B/A′ ≈ F in mod-B, (see Theorem 3.5), a property that does not extend to R, that is, in general R/A is not isomorphic to the left annihilator of A. (See Example 4.3(2) and Theorem 4.5.) Theorem 4.6 characterizes Frobenius rings among quasi-Frobenius (QF) rings. As an application of the main theorem, in Theorem 3.9 we prove that if A is generated as a right or left ideal by an idempotent e, then e is central (and R/A is then trivially right PF along with R). This generalizes the result of F. W. Anderson for quasi-Frobenius rings. (See Theorem 2.2 for a new proof.). In Proposition 1.6, we prove that a generalization of this result holds for finite products R of full matrix rings over local rings; namely, an ideal A is finitely generated as a right or left ideal iff A is generated by a central idempotent. We also note a theorem going back to Nakayama, Goursaud, and the author that every factor ring of R is right PF iff R is a uniserial ring. (See Theorem 5.1.).