A companion ideal of a multiplication module

Abstract

Let M _R be a faithful multiplication module, where R is a commutative ring. As defined by Anderson, $\theta \left( M \right) = \sum {_{x{\kern 1pt} \in M} \left[ {xR:M} \right]}$ this ideal has proved to be useful in studying multiplication modules. First of all a cancellation law involving M and the ideals contained in $\theta \left( M \right)$ is proved. Among various applications given, the following result is proved:: There exists a canonical isomorphism $\lambda$ from $Hom_R \left( {M,M} \right)$ onto $Hom_R \left( {\theta \left( M \right),\theta \left( M \right)} \right)$ such that for any σ ( Hom _R(M,M), x ( M, a ( θ (M), σ(xa) = x.(λ(σ)(a). As an application of this later result it is proved that M is quasi-injective if and only if θ(M) is quasi-injective.