BILINEAR PRODUCTS WITH A.C.C. ON ANNIHILATORS

ABSTRACT

Let be given two rings R and T, a bimodule with essential socle on both sides and such that U-duals of simples are of finite length, a left R-module , a right T-module and a non degenerate bilinear map ; we show that if either or satisfies both the A.C.C. and the D.C.C. on annihilators with respect to the map , then and are both of finite length. As an application, we generalize a theorem of Faith on finitely embedded commutative rings. Other situations in which the result is applicable are illustrated by a classic theorem of T. Lenagan.

Acknowledgment

It is for me both a honour and a pleasure to acknowledge the debt I owe to D. Herbera, who conjectured that the claim of Theorem 10 could be true, gave precious suggestions about its proof, and constantly supported me with very helpful discussions. In particular, the new proof of Lenagan's theorem given here is due to her. Moreover, I wish to express my thanks to the Department of Mathematics of the sUniversitat Autònoma de Barcelonas for their generous hospitality during the period in which this paper was written. Research supported by the Italian Consiglio Nazionale delle Ricerche through grant n. 201.01.130. Part of this paper was written while the author was visiting the Universitat Autònoma de Barcelona with partial financial support from DGESIC (Spain), through the grant PB98-0873.