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In this paper, we introduce and study the notions of $ M $-strongly hollow and $ M $-PS-hollow ideals where $ M $ is a module over a commutative ring $ R $. These notions are generalizations of strongly hollow ideals. We investigate some properties and characterizations of $ M $-strongly hollow ($ M $-PS-hollow) ideals. Then we define and study a topology on the set of all $ M $-PS-hollow ideals of a commutative ring $ R $. We investigate when this topological space is irreducible, Noetherian, $ T_{0} $, $ T_{1} $ and spectral space.

In this paper we introduce and study the concepts of graded second (gr-second) and graded coprimary (gr-coprimary) modules which are different from second and coprimary modules over arbitrary-graded rings. We list some properties and characterizations of gr-second and gr-coprimary modules and also study graded prime submodules of modules with gr-coprimary decompositions. We also deal with graded secondary representations for graded injective modules over commutative-graded rings. By using the concept of $$\sigma $$ -suspension $$(\sigma )M$$ of a graded module $$M,$$ we prove that a graded injective module over a commutative graded Noetherian ring has a graded secondary representation. [ABSTRACT FROM AUTHOR]