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.
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Al-Shaniafi, Y., Singh, S. A companion ideal of a multiplication module. Periodica Mathematica Hungarica 46, 1–8 (2003). https://doi.org/10.1023/A:1025727822090
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DOI: https://doi.org/10.1023/A:1025727822090