On Graded Primary Ideals (original) (raw)
Graded rings and essential ideals
Acta Mathematica Sinica, 1993
Let G be & group and A a G-graded ring. A (graded) ideal I of A is (graded) essential if I n J ~ 0 whenever J is a nonzero (graded) ideal of A. In this paper we study the relationship between graded essential ideals of A, essential ideals of the identity component Ae and essential ideals of the sna~h product A#G*. We apply our results to prime essential rings, irredundant subdirect sums and essentially nilpotent rings.
On Generalizations of Graded rrr-ideals
2021
In this article, we introduce a generalization of the concept of graded r-ideals in graded commutative rings with nonzero unity. Let G be a group, R be a G-graded commutative ring with nonzero unity and GI(R) be the set of all graded ideals of R. Suppose that φ : GI(R) → GI(R) ⋃ {∅} is a function. A proper graded ideal P of R is called a graded φ-r-ideal of R if whenever x, y are homogeneous elements of R such that xy ∈ P − φ(P ) and Ann(x) = {0}, then y ∈ P . Several properties of graded φ-r-ideals have been examined.
On graded primary-like submodules of graded modules over graded commutative rings
arXiv: Commutative Algebra, 2020
Let GGG be a group with identity eee. Let RRR be a GGG-graded commutative ring and MMM a graded RRR-module. In this paper, we introduce the concept of graded primary-like submodules as a new generalization of graded primary ideals and give some basic results about graded primary-like submodules of graded modules. Special attention has been paid, when graded submodules satisfies the gr-primeful property, to find extra properties of these graded submodules.
International Electronic Journal of Algebra, 2020
In this paper, we introduce the notion of graded Prüfer domain as a generalization of Prüfer domain to the graded case. We generalize several types of prime ideals associated to a module over a ring to the graded case and prove that most of them coincide over a graded Prüfer domain. Moreover, we investigate the graded primary decomposition of graded ideals in a graded Prüfer domain under certain conditions and give some applications of it.
GR-N-Ideals in Graded Commutative Rings
Acta Universitatis Sapientiae, Mathematica, 2019
Let G be a group with identity e and let R be a G-graded ring. In this paper, we introduce and study the concept of gr-n-ideals of R. We obtain many results concerning gr-n-ideals. Some characterizations of gr-n-ideals and their homogeneous components are given.
On Graded Semiprime and Graded Weakly Semiprime Ideals
2013
Let G be an arbitrary group with identity e and let R be a Ggraded ring. In this paper, we define graded semiprime ideals of a commutative G-graded ring with nonzero identity and we give a number of results concerning such ideals. Also, we extend some results of graded semiprime ideals to graded weakly semiprime ideals. Mathematics Subject Classification (2010):13A02, 13C05, 13A15
On graded hyperrings and graded hypermodules
2020
Let G be a monoid with identity e. In this paper, first we introduce the notions of G-graded hyperrings, graded hyperideals and graded hyperfields in the sense of Krasner hyperring R. Also, we define the notion of a greded R-hypermodules and some examples are presented. Then we investigate graded maximal, graded prime and graded primary hyperideals of a graded hyperring R. Finally, we study graded maximal, graded prime and graded primary subhypermodules of a graded R-hypermodule M and some interesting results on these concepts are given.
On Generalized k-Primary Rings
2015
Abstract. The present paper introduces and studies some new types of rings and ideals such as generalized k-primary rings ( resp. generalized k-primary ideals), principally generalized k-primary rings ( resp. principally generalized k-primary ideals) and completely generalized k-primary rings (resp. completely generalized k-primary ideals). Some properties of each are obtained and some characterizations of each type are given.
Graded Primal Submodules of Graded Modules
Journal of the Korean Mathematical Society, 2011
Let G be an abelian monoid with identity e. Let R be a G-graded commutative ring, and M a graded R-module. In this paper we first introduce the concept of graded primal submodules of M and give some basic results concerning this class of submodules. Then we characterize the graded primal ideals of the idealization R(+)M .