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Some notes on graded weakly 1-absorbing primary ideals

Alshehry Azzh Saad, Abu-Dawwas Rashid, Al-Rashdan Majd

Demonstratio Mathematica

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2023

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Vol. 56, nr 1

art. no. 20230111

EN

A proper graded ideal P of a commutative graded ring R is called graded weakly 1-absorbing primary if whenever x, y, z are nonunit homogeneous elements of R with 0≠xyz ∈ P , then either xy ∈ P or z is in the graded radical of P. In this article, we explore more results on graded weakly 1-absorbing primary ideals.

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Graded weakly 1-absorbing primary ideals

Bataineh Malik, Abu-Dawwas Rashid

Demonstratio Mathematica

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2023

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Vol. 56, nr 1

art. no. 20220214

EN

Let G be a group and R be a G-graded commutative ring with nonzero unity 1. In this article, we introduce the concept of graded weakly 1-absorbing primary ideals which is a generalization of graded 1-absorbing primary ideal. A proper graded ideal P of R is said to be a graded weakly 1-absorbing primary ideal of R if whenever nonunit elements x y z ∈ h(R), , such that 0 ≠ ∈ xyz ∈ P, then xy ∈ P or zn ∈ P , for some n ∈ N . Several properties of graded weakly 1-absorbing primary ideals are investigated.

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Graded I-second submodules

Bataineh Malik, Abu-Dawwas Rashid

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

1--8

EN

Let G be a group with identity e, R be a G-graded commutative ring with a nonzero unity 1, I be a graded ideal of R, and M be a G-graded R-module. In this article, we introduce the concept of graded I-second submodules of M as a generalization of graded second submodules of M and achieve some relevant outcomes.

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Almost graded multiplication and almost graded comultiplication modules

Bataineh Malik, Abu-Dawwas Rashid, Shtayat Jenan

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

325--331

EN

Let G be a group with identity e, R be a G-graded commutative ring with a nonzero unity 1 and M be a G-graded R-module. In this article, we introduce and study the concept of almost graded multiplication modules as a generalization of graded multiplication modules; a graded R-module M is said to be almost graded multiplication if whenever a ∈ h(R) satisfies AnnR (aM) = AnnR (M), then (0 :M a) = {0}. Also, we introduce and study the concept of almost graded comultiplication modules as a generalization of graded comultiplication modules; a graded R-module M is said to be almost graded comultiplication if whenever a ∈ h(R) satisfies AnnR (aM) = AnnR (M), then aM = M. We investigate several properties of these classes of graded modules.