Let G be a multiplicative group and R be a G-graded commutative ring and M a G-graded R-module. Various properties of multiplicative ideals in a graded ring are discussed and we extend this to graded modules over graded rings. We have also discussed the set of P-primary ideals and modules of R when P is a graded multiplication prime ideals and modules.
In this paper we prove that for a monoid S, products of indecomposable right S-acts are indecomposable if and only if S contains a right zero. Besides, we prove that subacts of indecomposable right S-acts are indecomposable if and only if S is left reversible. Ultimately, we prove that the one element right S-act ΘS is product flat if and only if S contains a left zero.
Let R be a commutative ring with identity. 2-absorbing ideals have been studied by A. Badawi. A proper ideal I of R is 2-absorbing if a, b, c ∈ R with abc ∈ I implies ab ∈ I or ac ∈ I or bc ∈ I. Let ϕ : I(R) → I(R) ∪ {∅} be a function where I(R) is the set of ideals of R. We call a proper ideal I of R a ϕ-2-absorbing ideal if a, b, c ∈ R with abc ∈ I − ϕ(I) implies ab ∈ I or ac ∈ I or bc ∈ I. So taking ϕ ∅ (J) = ∅ (resp., ϕ 0 (J) = 0, ϕ 2 (J) = J 2 ), a ϕ ∅ -2-absorbing ideal (resp., ϕ 0 -2-absorbing ideal, ϕ 2 -2-absorbing ideal) is a 2-absorbing ideal (resp., weakly 2-absorbing ideal, almost 2-absorbing ideal). We show that ϕ-2-absorbing ideals enjoy analogs of many of the properties of 2-absorbing ideals.
AMS Subject Classification: 13A15Key Words: almost 2-absorbing ideal, 2-absorbing ideal, ϕ-2-absorbing ideal, weakly 2-absorbing ideal * Throughout, R will be a commutative ring with identity. We denote the set of ideals of R by I(R). By a proper ideal I of R we mean an ideal I ∈ I(R) with I = R. We denote the set of proper ideals of R by I * (R).
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