Graded modules with Noetherian graded second spectrum

[摘要] Let $ R $ be a $ G $ graded commutative ring and $ M $ be a $ G $-graded $ R $-module. The set of all graded second submodules of $ M $ is denoted by $ Spec_G^s(M), $ and it is called the graded second spectrum of $ M $. We discuss graded rings with Noetherian graded prime spectrum. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. We also investigate $ Spec^s_G(M) $ with the Zariski topology from the viewpoint of being a Noetherian space.

[学科分类] 地球科学(综合)

[关键词] graded second submodule;graded second spectrum;graded Zariski socle;Zariski topology;Noetherian graded second spectrum