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IDEALS AND SUBMODULES OF MULTIPLICATION MODULES

  • LEE, SANG CHEOL (Department of Mathematics Education Chonbuk National University) ;
  • KIM, SUNAH (Department of Mathematics, Chosun University) ;
  • CHUNG, SANG-CHO (Department of Mathematics, Chungnam National University)
  • Published : 2005.09.01

Abstract

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))$\rightarrow$Spec$\_{R}$(M) and in particular, there exists a bijection: N(M)$\bigcap$Max(R)$\rightarrow$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 $\theta$(M) = $\sum$$\_{m(Rm :R M) of R has proved useful in studying multiplication modules. We generalize this ideal to prove the following result: Let R be a commutative ring with identity, P $\in$ Spec(R), and M a non-zero R-module satisfying (1) M is a finitely generated multiplication module, (2) PM is a multiplication module, and (3) P$^{n}$M$\neq$P$^{n+1}$ for every positive integer n, then $\bigcap$$^{$\_{n=1}$(P$^{n}$ + ann$\_{R}$(M)) $\in$ V(ann$\_{R}$(M)) = Supp(M) $\subseteq$ N(M).

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  1. Modules Satisfying the S-Noetherian Property and S-ACCR vol.44, pp.5, 2016, https://doi.org/10.1080/00927872.2015.1027377
  2. SOME PROPERTIES OF GR-MULTIPLICATION MODULES vol.20, pp.3, 2012, https://doi.org/10.11568/kjm.2012.20.3.315