On the ring of endomorphisms of a multiplication module

Abstract

Let R be a commutative ring with 1, and M is an R-module. Let E(M) = S be the ring of endomorphisms of M. It is known that if M is a multiplication module, then E(M) is commutative [6], and if M is a finitely generated multiplication module, then E(M) is isomorphic to R/ann (M), [4]. In this paper we give further results about E(M) for M a multiplication module.Let us call an endomorphism f of a module M a diagonal endomorphism if for all a E M, 3r E R, r depends on a, such that f(a) = ra. In Sec. 1 of this paper we show that every endomorphism f of a multiplication module is diagonal. In fact, we show that for each finite set al, a2,..., an E M, qr E R such that f(ai) = rai 1 < i < n, (see Theorem 1.5). On the other hand, we show that E(M) may not be isomorphic to R/ann (M) if M is not finitely generated. (See Example 1.) Motivated by the work in [9] and [3] on the ring of endomorphisms of a projective module, we study in Sec. 2 the Jacobson Radical of E(M). It turns out that J(E(M) has the same characterization for both projective modules and flat multiplication modules. Finally, we remark that for information on multiplication modules, see [1], [2], [5], [8]. w 1. The ring E(M) We start by the following: LEMMA 1.1. Let M be an R-module which is locally cyclic, and let f be an endomorphism of M. Then, for each a E M, 3ra E R ra depends on a, such that f(a) = raa.

Multiplication Modules and the Ideal

“…In [5] an example of a multiplication R-module M is given with R=0 : M = End R M. We end by illustrating this example.…”

“…Then End R M, the ring of R-endomorphisms of M, is a commutative ring with R=0 : M naturally embedded as a subring. Now for M ®nitely generated, R=0 : M End R M, but in general we may have R=0 : M = End R M (see [5] ). In Sec.…”