Generalizations of Prime Ideals

Let R be a commutative ring with identity. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R is weakly prime (resp., almost prime) if a, b ∈ R with ab ∈ I ? {0} (resp., ab ∈ I ? I ²) implies a ∈ I or b ∈ I. Let φ:?(R) → ?(R) ∪ {?} be a function where ?(R) is the set of ideals of R. We call a proper ideal I of R a φ-prime ideal if a, b ∈ R with ab ∈ I ? φ(I) implies a ∈ I or b ∈ I. So taking φ_?(J) = ? (resp., φ₀(J) = 0, φ₂(J) = J ²), a φ_?-prime ideal (resp., φ₀-prime ideal, φ₂-prime ideal) is a prime ideal (resp., weakly prime ideal, almost prime ideal). We show that φ-prime ideals enjoy analogs of many of the properties of prime ideals.     

On Generalizations of Prime Ideals

Let R be a commutative ring with identity. Let φ: S(R) → S(R) ∪ {?} be a function, where S(R) is the set of ideals of R. Suppose n ≥ 2 is a positive integer. A nonzero proper ideal I of R is called (n ? 1, n) ? φ-prime if, whenever a ₁, a ₂, ?, a _n  ∈ R and a ₁ a ₂?a _n  ∈ I?φ(I), the product of (n ? 1) of the a _i 's is in I. In this article, we study (n ? 1, n) ? φ-prime ideals (n ≥ 2). A number of results concerning (n ? 1, n) ? φ-prime ideals and examples of (n ? 1, n) ? φ-prime ideals are also given. Finally, rings with the property that for some φ, every proper ideal is (n ? 1, n) ? φ-prime, are characterized.     

MULTIPLICATIVELY CLOSED SETS OF IDEALS AND RESIDUAL DIVISION

Let R be a ring (commutative with identity), let Γ be a multiplicatively closed set of finitely generated nonzero ideals of R, for an ideal I of R let I _Γ = ∪ {I : G; G ∈ Γ}, and for an R-algebra A such that GA ≠ (0) for all G ∈ Γ let A ^Γ = ∪ {A : _T GA; G ∈ Γ}, where T is the total quotient ring of A. Then I _Γ is an ideal in R, I → I _Γ is a semi-prime operation (on the set I of ideals I of R) that satisfies a cancellation law, and it is a prime operation on I if and only if R = R ^Γ. Also, A ^Γ is an R-algebra and A → A ^Γ is a closure operation on any set A = {A; A is an R-algebra, R ? A, and if C is a ring between R and A, then regular elements in C remain regular in A}. Finally, several results are proved concerning relations between the ideals I _Γ and (IA)_ΓA and between the R-algebras R ^Γ and A ^Γ.

     

M-Canonical ideals in integral domains

We prove that a Priifer domain R has an m-canonical ideal J, that is, an ideal I such that J: (I: J) = J for every ideal J of R, if and only if R is h-local with only finitely many maximal ideals that are not finitely generated; moreover, if these conditions are satisfied, then the product of the non-finitely generated maximal ideals is an m-canonical ideal of R     

Filtrations,rees rings,and ideal transforms

Let I be an ideal, and let f = {K_n|n ≥ 0 } be a filtration of the Noetherian ring R, such that Iⁿ ⊆ K_n for all n ≥ 0. We study when the Rees ring $\text{R}$(f) is either finite or integral over the Rees ring $\text{R}$(I), for two types of filtrations f which have recently drawn interest. If I and J are ideals in R, and if m(n) is the least power of J such that (Iⁿ : J^{m(n) + 1}), we show that the function m(n) is eventually non-decreasing. For J regular, we characterize when it is eventually constant.     

Locally Invariant and Semi-Invariant Right Cones

A right cone of a group G is a submonoid H of G so that for a, b ∈ H either aH ? bH or bH ? aH and G = {ab ^?1 | a, b ∈ H}. Valuation rings, right chain rings, the cones of right ordered groups provide examples. It is proved, see Theorem 17, that a semi-invariant right cone H with d.c.c. for prime ideals satisfies Ha ? aH for all a ∈ H, that is H is right invariant. Essential is the following Theorem 9: Let H be a locally invariant right cone in G with d.c.c. for prime ideals, and let I ≠ H be an ideal in H. Then P _l (I) ? P _r (I) for the associated left and right prime ideals of I.     

On simple factorization of invertible matrices

Let I be an ideal of a ring R. We say that R is a generalized I-stable ring provided that aR+bR=R with a?∈?1+I,b?∈?R implies that there exists a y?∈?R such that a+by?∈?K(R), where K(R)={x?∈?R?∣?? s, t?∈?R such that sxt=1}. Let R be a generalized I-stable ring. Then every A?∈?GL_n (I) is the product of 13n?12 simple matrices. Furthermore, we prove that A is the product of n simple matrices if I has stable rank one. This generalizes the results of Vaserstein and Wheland on rings having stable rank one.     

ON THE NORMALITY OF MONOMIAL IDEALS OF MIXED PRODUCTS

Let R = K[x, y] be a polynomial ring in two disjoint sets of variables x, y over a field K. We study ideals of mixed products L = I_kJ_r + I_sJ_t such that k + r = s + t, where I_k (resp. J_r ) denotes the ideal of R generated by the square-free monomials of degree k (resp. r) in the x (resp. y ) variables. Our main result is a characterization of when a given ideal L of mixed products is normal.

     

A SURJECTIVE HOMOMORPHISM OF LOCAL COHOMOLOGY MODULES

Let R be a local ring and M a finitely generated generalized Cohen-Macaulay R-module such that dim _R M = dim _R M/αM + height_Mα a for all ideals α of R. Suppose that H_I ^j(M) ≠ 0 for an ideal I of R and an integer j > height_M I. We show that there exists an ideal J ? such that a. height_M J = j;

b. the natural homomorphism H_I ^j(M) → H_I ^j(M) is an isomorphism, for all i > j; and,

c. the natural homomorphism H_I ^j(M) → H_I ^j(M) is surjective.

By using this theorem, we obtain some results about Betti numbers, coassociated primes, and support of local cohomology modules.     

On the Regularity of Operations of Ideals

Let I be a homogeneous ideal of a polynomial ring K[x₁,…, x_n] over a field K, and denote the Castelnuovo–Mumford regularity of I by reg(I). When I is a monomial complete intersection, it is proved that reg(I^m) ≤ mreg(I) holds for any m ≥ 1. When n = 3, for any homogeneous ideals I and J of K[x₁, x₂, x₃], one has that reg(I ? J), reg(IJ) and reg(I ∩ J) are all upper bounded by reg(I) +reg(J), while reg(I + J) ≤reg(I) +reg(J) ?1.     

Uppers to zero in R[x] and almost principal ideals

Let R be an integral domain with quotient field K and f(x) a polynomial of positive degree in K[x]. In this paper we develop a method for studying almost principal uppers to zero ideals. More precisely, we prove that uppers to zero divisorial ideals of the form I = f(x)K[x] ∩ R[x] are almost principal in the following two cases:

• J, the ideal generated by the leading coefficients of I, satisfies J ^?1 = R.
• I ^?1 as the R[x]-submodule of K(x) is of finite type.

Furthermore we prove that for I = f(x)K[x] ∩ R[x] we have:

• I ^?1 ∩ K[x] = (I: K(x) I).
• If there exists p/q ∈ I ^?1 ? K[x], then (q, f) ≠ 1 in K[x]. If in addition q is irreducible and I is almost principal, then I′ = q(x)K[x] ∩ R[x] is an almost principal upper to zero.

Finally we show that a Schreier domain R is a greatest common divisor domain if and only if every upper to zero in R[x] contains a primitive polynomial.     

Strongly irreducible ideals of a commutative ring

An ideal I of a ring R is said to be strongly irreducible if for ideals J and K of R, the inclusion J∩K⊆I implies that either J⊆I or K⊆I. The relationship among the families of irreducible ideals, strongly irreducible ideals, and prime ideals of a commutative ring R is considered, and a characterization is given of the Noetherian rings which contain a non-prime strongly irreducible ideal.     

Ideals in tensor powers of the enveloping algebra u(sl 2)

Let U = U(_sl₂)^?n be the tensor power of n copies of the enveloping algebra U(sl ₂) over an arbitrary field K of characteristic zero. In this paper we list the prime ideals of U by generators and classify them by height. If Z is the center of U and J is a prime ideal of Z, there are exactly 2₅ prime ideals I of U with I ∩ Z = J, where 0 ≤ s = s(J) ≤ n is an integer. Indeed, with respect to inclusion, they form a lattice isornorphic to the lattice of subsets of a set. When J is a maximal ideal of Z, there are only finitely many two-sided ideals of U containing J, They are presented by generators and their lattice is described, In particular, for each such J there exists a unique maximal ideal of U containing J and a unique ideal of U minimal with respect to the property that it properly contains JU. Similar results are given in the case when U is the tensor product of infinitely many copies of U(sl ₂).     

The Cohen–Macaulay and Gorenstein Properties of Rings Associated to Filtrations

Let (R, m) be a Cohen–Macaulay local ring, and let ? = {F _i } _i∈? be an F ₁-good filtration of ideals in R. If F ₁ is m-primary we obtain sufficient conditions in order that the associated graded ring G(?) be Cohen–Macaulay. In the case where R is Gorenstein, we use the Cohen–Macaulay result to establish necessary and sufficient conditions for G(?) to be Gorenstein. We apply this result to the integral closure filtration ? associated to a monomial parameter ideal of a polynomial ring to give necessary and sufficient conditions for G(?) to be Gorenstein. Let (R, m) be a Gorenstein local ring, and let F ₁ be an ideal with ht(F ₁) = g > 0. If there exists a reduction J of ? with μ(J) = g and reduction number u: = r _J (?), we prove that the extended Rees algebra R′(?) is quasi-Gorenstein with a-invariant b if and only if J ⁿ : F _u  = F _n+b?u+g?1 for every n ∈ ?. Furthermore, if G(?) is Cohen–Macaulay, then the maximal degree of a homogeneous minimal generator of the canonical module ω _G(?) is at most g and that of the canonical module ω _R′(?) is at most g ? 1; moreover, R′(?) is Gorenstein if and only if J ^u : F _u  = F _u . We illustrate with various examples cases where G(?) is or is not Gorenstein.     

Resolutions of subsets of finite sets of points in projective space

Abstract

Eisenbud et al. proved a number of results regarding Gröbner bases and initial ideals of those ideals J in the free associative algebra K ?X ₁,…, X _n ? which contain the commutator ideal. We prove similar results for ideals which contains the anti-commutator ideal (the defining ideal of the exterior algebra). We define one weak notion of generic initial ideals in K ?X ₁,…, X _n ?, and show that generic initial ideals of ideals containing the anti-commutator ideal, or the commutator ideal, are finitely generated.     

The Hilbert–Kunz Function in Graded Dimension Two

Let R denote a two-dimensional normal standard-graded K-domain over the algebraic closure K of a finite field of characteristic p, and let I ? R denote a homogeneous R ₊-primary ideal. We prove that the Hilbert–Kunz function of I has the form ? (q) = e _HK (I)q ² + γ(q) with rational Hilbert–Kunz multiplicity e _HK (I) and an eventually periodic function γ(q).     

The Inclusion Ideal Graph of Rings

Let R be a ring with unity. The inclusion ideal graph of a ring R, denoted by In(R), is a graph whose vertices are all nontrivial left ideals of R and two distinct left ideals I and J are adjacent if and only if I ? J or J ? I. In this paper, we show that In(R) is not connected if and only if R ? M ₂(D) or D ₁ × D ₂, for some division rings, D, D ₁ and D ₂. Moreover, we prove that if In(R) is connected, then diam(In(R)) ≤3. It is shown that if In(R) is a tree, then In(R) is a caterpillar with diam(In(R)) ≤3. Also, we prove that the girth of In(R) belongs to the set {3, 6, ∞}. Finally, we determine the clique number and the chromatic number of the inclusion ideal graph for some classes of rings.     

Ideals of Some Matrix Rings

For a commutative ring K the conception of a strongly maximal ideal J was introduced by Kuzucuoglu and Levchuk in 2000. Denote by R_n(K,J) the ring of all n×n-matrices over K with elements from J on and above the main diagonal. Recent results on ideals of the ring R_n(K,J) for this case, ideals of the associated Lie ring and normal subgroups of the adjoint group are considered in this paper. Also ideals of R_n(K,J) for the case of an arbitrary associative ring K with the identity are investigated.