Graded primal ideals (original) (raw)

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.

On Graded Primary Ideals

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 purely-prime ideals with applications

Communications in Algebra

In this paper, new algebraic and topological results on purely-prime ideals of a commutative ring are obtained. Some applications of this study are also given. In particular, the new notion of semi-noetherian ring is introduced and Cohen type theorem is proved.

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.

Chains of Prime Ideals and Primitivity of ℤ mathbbZ\mathbb {Z}mathbbZ -Graded Algebras

Algebras and Representation Theory, 2015

In this paper we provide some results regarding affine, prime, Z-graded algebras R = ⊕ i∈Z R i generated by elements with degrees 1, −1 and 0, with R 0 finite-dimensional. The results are as follows. These algebras have a classical Krull dimension when they have quadratic growth. If R k ̸ = 0 for almost all k then R is semiprimitive. If in addition R has GK dimension less than 3 then R is either primitive or PI. The tensor product of an arbitrary Brown-McCoy radical algebra of Gelfand-Kirillov dimension less than three and any other algebra is Brown-McCoy radical. Keywords Graded algebras • primitive rings • semiprimitive rings • Brown-McCoy radical • chains of prime ideals • GK dimension • growth of algebra Mathematics Subject Classification (2000) 16W50 • 16P90 A.

Associated graded rings of one-dimensional analytically irreducible rings II

Journal of Algebra, 2011

Lance Bryant noticed in his thesis [3], that there was a flaw in our paper [2]. It can be fixed by adding a condition, called the BF condition in [3]. We discuss some equivalent conditions, and show that they are fulfilled for some classes of rings, in particular for our motivating example of semigroup rings. Furthermore we discuss the connection to a similar result, stated in more generality, by Cortadella-Zarzuela in [4]. Finally we use our result to conclude when a semigroup ring in embedding dimension at most three has an associated graded which is a complete intersection. 2000 Mathematics Subject Classification: 13A30 If x ∈ R is an element of smallest positive value, i.e. v(x) = e, then xR is a minimal reduction of the maximal ideal, i.e. m n+1 = xm n , for n >> 0. Conversely each minimal reduction of the maximal ideal is a principal ideal generated by an element x of value e. The smallest integer n such that m n+1 = xm n is called the reduction number and we denote it by r. Observe that, if v(x) = e, then Ap e (S) = S \(e+S) = v(R)\v(xR), therefore w j / ∈ v(xR), for j = 0,. .. , e − 1. Consider the m-adic filtration m ⊃ m 2 ⊃ m 3 ⊃. .. . If a ∈ R, we set ord(a) := max{i | a ∈ m i }. If s ∈ S, we consider the semigroup filtration v(m) ⊃ v(m 2) ⊃. .. and set vord(s) := max{i | s ∈ v(m i)}. If a ∈ m i , then v(a) ∈ v(m i) and so ord(a) ≤ vord(v(a)). According to [3], we say that the m-adic filtration is essentially divisible with respect to the minimal reduction xR if, whenever u ∈ v(xR), then there is an a ∈ xR with v(a) = u and ord(a) = vord(u). The m-adic filtration is essentially divisible if there exists a minimal reduction xR such that it is essentially divisible with respect to xR. We fix for all the paper the following notation. Set, for j = 0,. .. , e − 1, b j = max{i|w j ∈ v(m i)}, and let c j = max{i|w j ∈ v(m i + xR)}. Note that the numbers b j 's do not depend on the minimal reduction xR, on the contrary the c j 's depend on xR. Lemma 1.1 If I and J are ideals of R, then v(I +J) = v(I)∪v(J) is equivalent to v(I ∩ J) = v(I) ∩ v(J).