Malaysian Journal of Mathematical Sciences, May 2017, Vol. 11, No. 2
Graded Modules Over First Strongly Graded Rings
Fida, M. and Maisa, K.
Corresponding Email: f.mohammad@psut.edu.jo
Received date: 20 October 2016
Accepted date: 12 April 2017
Abstract:
Let \(G\) be a group with identity \(e\) and \(R = \underset{g \in G}{\bigoplus} R_g\) be a \(G\)-graded ring. We say \(R\) is first strongly graded if \(R_gR_h=R_{gh}\), for every \(g, h \in supp(R,G)\). In this paper, we investigate several properties of group-graded modules
over first strongly graded rings concerning annihilators and multiplication modules, semisimple modules, and other aspects of modules. For
example, we prove that the nonzero components of a graded module over a commutative first strongly graded ring whose degrees lie in \(supp(R,G)\) possess an equal annihilators over \(R\). Adding an extra condition, we can further partition the group \(G\) into three parts such that the annihilators of the components of the graded module whose degrees lying in the same part possess the same annihilator. Also, we show that over first strongly graded rings the \(gr\)-semisimple \(R\)-modules are only the flexible \(R\)-modules with semisimple \(e\)-components.
Keywords: first strongly graded rings, graded modules, flexible modules, first strongly graded modules, semisimple modules, annihilator
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