IJPAM: Volume 76, No. 2 (2012)

GRADED PRIME SUBMODULES
OVER MULTIPLICATION MODULES

Malik Bataineh$^1$, Ala' Lutfi Khazaaleh$^2$
$^{1,2}$Department of Mathematics and Statistics
Jordan University of Science and Technology
Irbid, 22110, JORDAN


Abstract. Let $G$ be an abelian group with identity $e$, $R$ be a $G-$graded commutative ring and $M$ a graded $R-$module where all modules are unital. Various generalizations of graded prime ideals have been studied. For example, a proper graded ideal $I$ is a graded weakly (resp., almost) prime ideal if $0\neq ab\in I$ (resp., $ab\in I-I^{2}$) implies $% a\in I$ or $b\in I$. Throughout this work, we define that a proper graded submodule $N$ of $M$ is a graded almost prime if $am\in N-(N:M)N$ implies $a\in (N:M)$ or $m\in N $. We show that graded almost prime submodules enjoy analogs of many of the properties of prime submodules.

Received: November 12, 2011

AMS Subject Classification: 13A02, 16W50

Key Words and Phrases: graded rings, graded multiplication modules, graded almost prime submodules

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 76
Issue: 2