Flat modules and multiplication modules

Abstract

I.& R be a commutative ring with an identity I, and let M be a unitary (right) R-module. Recall that trace M = F(M) = zZ(M) where the sum runs over allflinear maps 1 E M*, M* being the dual of M. It is known that if M is a finitely generated multiplication module, then T(M) = D(M) = annann (M) [8], [9]. However, if M is an arbitrary multiplication module, then T(M) # D(M) in general. One of the aims of this paper is to study Z’(M). For each a E M, let D, = annann (a), and put D,(M) = = 2 D,. It turns out that if M is a multiplication module then T(M) = UEM = D,(M) (Th. 2.4). In 8 3, various conditions are studied that guarantee the finiteness of a multiplication module. For example, if T(M) is finitely generated, then M is finitely generated (Th. 3.4). In Q 4, we look for conditions under which a multiplication module becomes a flat module. It is known that a finitely generated multiplication ideal with a pure annihilator is a flat ideal [6]. In Th. 4.1, we generalize this result to orbitrary multiplication modules with pure annihilators. Finally, we remark that all rings considered in this paper are commutative with 1.