Graded primal ideals (original) (raw)
On Graded Weakly S-Prime Ideals
2021
Let R be a commutative graded ring with unity, S be a multiplicative subset of homogeneous elements of R and P be a graded ideal of R such that P\bigcap S=\emptyset In this article, we introduce several results concerning graded S-prime ideals. Then we introduce the concept of graded weakly S-prime ideals which is a generalization of graded weakly prime ideals. We say that P is a graded weakly S-prime ideal of R if there exists s\in S such that for all x, y\in h(R), if 0\neq xy\in P, then sx\in P or sy\in P. We show that graded weakly S-prime ideals have many acquaintance properties to these of graded weakly prime ideals.
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.
Let G be a group and R be a G-graded commutative ring, i.e., R = ⊕ g∈G Rg and Rg Rh ⊆Rgh for all g, h ∈G. In this paper, we study the graded primary ideals and graded primary G-decomposition of a graded ideal.
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 .
On a certain transitivity of the graded ring associated with an ideal
Proceedings of the American Mathematical Society, 1982
A simple but useful result will be given concerning a certain transitivity of the property that the graded ring associated with an ideal is a domain. As a consequence, we compute the graded rings associated with the defining prime ideals of certain determinantal varieties or of their projections from infinity to a hyperplane and get two new classes of primes having the equality of ordinary and symbolic powers in polynomial rings over a field.
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
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.
Graded Rings in Which Every Proper Graded Ideal is Almost GR-Prime
2009
In this paper, we study further properties of almost and n-almost gr-prime ideals in a graded ring R. In particular, we investigate some conditions under which a graded ideal is almost gr-prime. Finally, we give a characterization for graded rings in which every proper graded ideal is almost gr-prime.
Journal of the Mathematical Society of Japan, 1978
In this paper, we study a Noetherian graded ring RRR and the category of graded R-modules. We consider injective objects of this category and we define the graded Cousin complex of a graded R-module MMM . These concepts are essential in this paper