M-cancellation Ideals

07/08/2020

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by   Peyman Nasehpour, Ph.D., et al.

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Let R be a commutative ring with non–zero identy and let M be an R–module. An ideal a of R is called an M-cancellation ideal if whenever aP = aQ for submodules P and Q of M, then P = Q. This notion is a generalization of the notion, cancellation ideal. We use M-cancellation ideals and a generalization of Dedekind–Mertens lemma to prove that for an R–module M with ZR(M) = {0}, the following statements are equivalent : (i) Every non–zero finitely generated ideal of R is an M-cancellation ideal of R. (ii) For every f ∈ R[t] and g ∈ M[t], c(fg) = c(f)c(g).