IDEALS AND SUBMODULES OF MULTIPLICATION MODULES - LEE, SANG CHEOL; KIM, SUNAH; CHUNG, SANG-CHO (Journal of the Korean Mathematical Society, 2005)
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)$\bigcap$V(ann$_{R}$(M))$→$Spec$_{R}$(M) and in particular, there exists a bijection: N(M)$\bigcap$Max(R)$→$Max$_{R}$(M), (2) N(M) $\bigcap$ V(ann$_{R}$(M)) = Supp(M) $\bigcap$ V(ann$_{R}$(M)), and (3) for every ideal I of R, The ideal $θ$(M) = $\sum$$_{m(Rm :R M) of R has proved useful in studying multiplication modules. We generalize this ideal to prove the following result: Let R be a commutative ring with identity, P $\in$ Spec(R), and M a non-zero R-module satisfying (1) M is a finitely generated multiplication module, (2) PM is a multiplication module, and (3) P$^{n}$M$≠$P$^{n+1}$ for every positive integer n, then $\bigcap$$^{$_{n=1}$(P$^{n}$ + ann$_{R}$(M)) $\in$ V(ann$_{R}$(M)) = Supp(M) $\subseteq$ N(M).