Quasi-Frobenius extensions with Morita duality

Abstract

A duality theory for modules was established by Morita [8] and Azumaya [2]. In [lo], Miiller characterized rings with duality using the concept of linear compactness defined by Zelinsky [ 131. Rings with duality are closely connected with Frobenius algebras introduced by Nakayama [ 111 and their generalization, quasi-Frobenius (qF) rings. Recently Vamos [ 121 studied the inheritance of duality between a ring and its over ring, and showed that for a ring extension A 3 B such that A, is finitely generated by elements which centralize B, if B has a duality induced by U,, then A has a duality induced by Hom,(A, U), . While, for ring extensions, Kasch [4] introduced the notion of Frobenius extensions as a generalization of that of Frobenius algebras, and Mtiller [9] extended it to the notion of qF extensions. In this paper we shall extend the Vamos’ result above to the case where A XI B satisfies the condition that there exist a, ,..., a, in A such that A = xi aiB and a,B = Ba, (i = l,..., n), and study Frobenius or qF extensions with duality. Our main result is stated as follows: Let A/B be a Frobenius (resp. qF) extension satisfying the above condition. If ,,U, defines a Morita duality, then rHomg(A, U), defines a Morita duality and T/A is a Frobenius (resp. qF) extension, where r= End,(Hom,(A, U)). If 1V, defines a Morita duality, then qVB defines a Morita duality and Q/Z: is a Frobenius (resp. qF) extension, where Q = End,( v>. The present paper consists of three sections. The first section is concerned with results on bimodules which will be employed in the sequel. The second section deals with Frobenius or qF extensions with Morita duality. The third section is devoted to give an application of our results to studying the inheritance of Morita duality between a ring A and a fixed subring AG with a finite group G of ring automorphisms of A. Throughout the present paper, all rings have a 1, which acts unitally and is preserved by homomorphisms and subrings. Module homomorphisms will be written on the side opposite the scalars. 275 0021~8693/81/120275-12$02.00/0