IDEALS AND SUBMODULES OF MULTIPLICATION MODULES LEE, SANG CHEOL; KIM, SUNAH; CHUNG, SANG-CHO;
Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for every submodule N of M there exists an ideal I of R such that N = 1M. Let M be a non-zero multiplication R-module. Then we prove the following: (1) there exists a bijection: N(M)V(ann(M))Spec(M) and in particular, there exists a bijection: N(M)Max(R)Max(M), (2) N(M) V(ann(M)) = Supp(M) V(ann(M)), and (3) for every ideal I of R, The ideal (M) = (Rm :R M) of R has proved useful in studying multiplication modules. We generalize this ideal to prove the following result: Let R be a commutative ring with identity, P Spec(R), and M a non-zero R-module satisfying (1) M is a finitely generated multiplication module, (2) PM is a multiplication module, and (3) PMP for every positive integer n, then (P + ann(M)) V(ann(M)) = Supp(M) N(M).
prime submodules;maximal submodulesq;finitely generated modules;multiplication modules;