On Some Characteristic Properties of Quasi-frobenius and Regular Rings

Recently the first writer [l] gave a characterization of quasiFrobenius rings, introduced formerly by the second writer [3], in terms of a condition proposed by K. Shoda, which reads: A ring A satisfying minimum condition and possessing a unit element is a quasi-Frobenius ring if and only if A satisfies the following condition:1 (a) every (A -left-) homomorphism of a left-ideal of A into A may be given by the right multiplication of an element of A. In the present note we shall offer a simpler2 proof of this, making use of the second writer's former characterization of quasi-Frobenius rings and a theorem in a previous joint note of the writers. The present approach starts, contrary to the one in [l],3 with a theorem (Theorem 1) which is independent of any chain condition (or is concerned with maximum condition at most (corollary to Theorem 1) and which is perhaps of interest by itself. We shall also show that a remark at the end of [l], concerning semisimple rings with minimum condition, may be freed from chain condition, to yield a certain characterization of von Neumann's regular rings (Theorem 3).