DOI QR코드

DOI QR Code

SEMIPRIME SUBMODULES OF GRADED MULTIPLICATION MODULES

  • Lee, Sang-Cheol (Department of Mathematics Education Chonbuk National University) ;
  • Varmazyar, Rezvan (Department of Mathematics Islamic Azad University)
  • Received : 2011.04.27
  • Published : 2012.03.01

Abstract

Let G be a group. Let R be a G-graded commutative ring with identity and M be a G-graded multiplication module over R. A proper graded submodule Q of M is semiprime if whenever $I^nK{\subseteq}Q$, where $I{\subseteq}h(R)$, n is a positive integer, and $K{\subseteq}h(M)$, then $IK{\subseteq}Q$. We characterize semiprime submodules of M. For example, we show that a proper graded submodule Q of M is semiprime if and only if grad$(Q){\cap}h(M)=Q+{\cap}h(M)$. Furthermore if M is finitely generated then we prove that every proper graded submodule of M is contained in a graded semiprime submodule of M. A proper graded submodule Q of M is said to be almost semiprime if (grad(Q)$\cap$h(M))n(grad$(0_M){\cap}h(M)$) = (Q$\cap$h(M))n(grad$(0_M){\cap}Q{\cap}h(M)$). Let K, Q be graded submodules of M. If K and Q are almost semiprime in M such that Q + K $\neq$ M and $Q{\cap}K{\subseteq}M_g$ for all $g{\in}G$, then we prove that Q + K is almost semiprime in M.

Keywords

graded multiplication module;semiprime submodule;almost semiprime

References

  1. R. Ameri, On the prime submodules of multiplication modules, Int. J. Math. Math. Sci. 2003 (2003), no. 27, 1715-1724. https://doi.org/10.1155/S0161171203202180
  2. S. E. Atani, On graded prime submodules, Chiang Mai J. Sci. 33 (2006), no. 1, 3-7.
  3. S. E. Atani and F. Farzalipour, On graded secondary modules, Turkish J. Math. 31 (2007), no. 4, 371-378.
  4. A. Barnard, Multiplication modules, J. Algebra 71 (1981), no. 1, 174-178. https://doi.org/10.1016/0021-8693(81)90112-5
  5. K. H. Oral, U. Tekir, and A. G. Agargun, On graded prime and primary submodules, Turk J. Math. 35 (2011), 159-167.
  6. P. F. Smith, Some remarks on multiplication module, Arch. Math. (Basel) 50 (1988), no. 3, 223-235. https://doi.org/10.1007/BF01187738

Cited by

  1. Graded semiprime submodules and graded semi-radical of graded submodules in graded modules 2016, https://doi.org/10.1007/s11587-016-0312-x