2021/02/08 by Azzh Saad Alshehry, Alshehry, Azzh Saad, Rashid Abu-Dawwas +3 · 1 citation
Mathematics · #Rings, Modules, and Algebras #Advanced Topics in Algebra #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.2102.04155
Let R be a graded commutative ring with non-zero unity 1 and M be a graded unitary R-module. Let GS(M) be the set of all graded R-submodules of M and ϕ: GS(M)→ GS(M)\bigcup\∅\ be a function. A proper graded R-submodule K of M is said to be a graded ϕ-prime R-submodule of M if whenever r is a homogeneous element of R and m is a homogeneous element of M such that rm∈ K-ϕ(K), then either m∈ K or r∈ (K:RM). If ϕ(K)=∅ for all K∈ GS(M), then a graded ϕ-prime submodule is exactly a graded prime submodule. If ϕ(K)=\0\ for all K∈ GS(M), then a graded ϕ-prime submodule is exactly a graded weakly prime submodule. Several properties of graded ϕ-prime submodules have been investigated.