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.     

The Leading Ideal of a Complete Intersection of Height Two in a 2-Dimensional Regular Local Ring

Let (S,𝔫) be a 2-dimensional regular local ring and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. Let I* be the leading ideal of I in the associated graded ring gr𝔫(S), and set R = S/I and 𝔪 = 𝔫/I. In Goto et al. (2007 Goto , S. , Heinzer , W. , Kim , M.-K. ( 2007 ). The leading ideal of a complete intersection of height two, II . J. Algebra 312 : 709 – 732 . [Google Scholar]), we prove that if μ _G (I*) = n, then I* contains a homogeneous system {ξ _i }_1≤i≤n of generators such that deg ξ _i  + 2 ≤ deg ξ _i+1 for 2 ≤ i ≤ n ? 1, and ht _G (ξ₁, ξ₂,…, ξ _n?1) = 1, and we describe precisely the Hilbert series H(gr𝔪(R), λ) in terms of the degrees c _i of the ξ _i and the integers d _i , where d _i is the degree of D _i  = GCD(ξ₁,…, ξ _i ). To the complete intersection ideal I = (f, g)S we associate a positive integer n with 2 ≤ n ≤ c ₁ + 1, an ascending sequence of positive integers (c ₁, c ₂,…, c _n ), and a descending sequence of integers (d ₁ = c ₁, d ₂,…, d _n  = 0) such that c _i+1 ? c _i  > d _i?1 ? d _i  > 0 for each i with 2 ≤ i ≤ n ? 1. We establish here that this necessary condition is also sufficient for there to exist a complete intersection ideal I = (f, g) whose leading ideal has these invariants. We give several examples to illustrate our theorems.     

Power Values of Derivations with Annihilator Conditions on Lie Ideals in Prime Rings

Let R be a prime ring of char R ≠ 2, d a nonzero derivation of R, U a noncentral Lie ideal of R, and a ∈ R. If au ^{n ₁} d(u) ^{n ₂} u ^{n ₃} d(u) ^{n ₄} u ^{n ₅} … d(u) ^{n _k?1} u ^{n _k}  = 0 for all u ∈ U, where n ₁, n ₂,…,n _k are fixed non-negative integers not all zero, then a = 0 and if a(u ^s d(u)u ^t ) ⁿ  ∈ Z(R) for all u ∈ U, where s ≥ 0, t ≥ 0, n ≥ 1 are some fixed integers, then either a = 0 or R satisfies S ₄, the standard identity in four variables.     

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.     

The Leading Ideal of a Complete Intersection of Height Two,Part III

Let (S, 𝔫) be an s-dimensional regular local ring with s > 2, and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. As in [2 Goto , S. , Heinzer , W. , Kim , M.-K. ( 2006 ). The leading ideal of a complete intersection of height two . J. Algebra 298 : 238 – 247 . [Google Scholar], 3 Goto , S. , Heinzer , W. , Kim , M.-K. ( 2007 ). The leading ideal of a complete intersection of height two, II . J. Algebra 312 : 709 – 732 . [Google Scholar]], we examine the leading form ideal I* of I in the associated graded ring G: = gr_𝔫(S). Let μ _G (I*) = n ≥ 3, and let {ξ₁, ξ₂,…, ξ _n } be a minimal homogeneous system of generators of I* such that ξ₁ = f* and ξ₂ = g*, and c _i : = deg ξ _i  ≤ deg ξ _i+1: = c _i+1 for each i ≤ n ? 1. For m ≤ n, we say that K _m : = (ξ₁,…, ξ _m )G is an ideal generated by part of a minimal homogeneous generating set of I*. Let D _i : = GCD(ξ₁,…, ξ _i ) and d _i  = deg D _i for i with 1 ≤ i ≤ m. Let K _m be perfect with ht _G K _m  = 2. We prove that the following are equivalent: 1. deg ξ _i+1 = deg ξ _i  + (d _i?1 ? d _i ) +1, for all i with 3 ≤ i ≤ m ? 1;

2. deg ξ _i+1 ≤ deg ξ _i  + (d _i?1 ? d _i ) +1, for all i with 3 ≤ i ≤ m ? 1.

Furthermore, if these equivalent conditions hold, then K _m  = I*. Moreover, if e(G/K _m ) = e(G/I*), we prove that K _m  = I*. We illustrate with several examples in the cases where K _m is or is not perfect.