The Zariski topology on the graded classical prime spectrum of a graded module over a graded commutative ring. (English) Zbl 1474.13002

Al-Zoubi, Khaldoun; Jaradat, Malik

Mat. Vesn. 70, No. 4, 303-313 (2018).

Summary: Let \(G\) be a group with identity \(e\). Let \(R\) be a \(G\)-graded commutative ring and \(M\) a graded \(R\)-module. A proper graded submodule \(N\) of \(M\) is called a graded classical prime if whenever \(r,s\in h(R)\) and \(m\in h(M)\) with \(rsm\in N\), then either \(rm\in N\) or \(sm\in N\). The graded classical prime spectrum \(\mathrm{Cl.Spec}^g(M)\) is defined to be the set of all graded classical prime submodules of \(M\). In this paper, we introduce and study a topology on \(\mathrm{Cl.Spec}^g(M)\), which generalizes the Zariski topology of graded ring \(R\) to graded module \(M\), called Zariski topology of \(M\), and investigate several properties of the topology.

MSC:

13A02

Graded rings

Keywords:

graded classical prime spectrum; graded classical prime submodule; Zariski topology

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