A Characterization of Quasi-Frobenius Rings

In this note we shall consider the problem: in what ring A can every homomorphism between two left ideals be extended to a homo-morphism of A? ( " Homomorphism " means " operator homomorphism "). We shall call this condition as Shoda's condition.1 * When A is a ring with a unit element, Shoda's condition is equivalent to the next one: (α): every homomorphism between two left ideals is given by the right multiplication of an element of A. The main purpose of this note is to show that if A is a ring with a unit element satisfying the minimum condition for left and right ideals, then A satisfies Shoda's condition if and only if A is a quasi-Frobenius ring. T. Nakayama characterized quasi-Frobenius rings as the rings in which the duality relations Z(r (!.))= = I and r(Z(x)) = r hold for every left ideal I and right ideal x.2) Our result gives another characterization of quasi-Frobenius rings. A denotes always a ring with the minimum condition for left and right ideals. Let N be the radical of A and A = A/N — Ά τ + •• • + A n be the direct decomposition of A into simple two-sided ideals. Then, as is well known, we have two direct decompositions of A: A = Σ Σ Ae κ,, + l(E} = Σ §k, *A+r (#) (1) κ=l t=ι κ=l i=