Depth of Initial Ideals of Normal Edge Rings

Let G be a finite graph on the vertex set [d] = {1,…, d} with the edges e ₁,…, e _n and K[t] = K[t ₁,…, t _d ] the polynomial ring in d variables over a field K. The edge ring of G is the semigroup ring K[G] which is generated by those monomials t ^e  = t _i t _j such that e = {i, j} is an edge of G. Let K[x] = K[x ₁,…, x _n ] be the polynomial ring in n variables over K, and define the surjective homomorphism π: K[x] → K[G] by setting π(x _i ) = t ^{e _i} for i = 1,…, n. The toric ideal I _G of G is the kernel of π. It will be proved that, given integers f and d with 6 ≤ f ≤ d, there exists a finite connected nonbipartite graph G on [d] together with a reverse lexicographic order <_rev on K[x] and a lexicographic order <_lex on K[x] such that (i) K[G] is normal with Krull-dim K[G] = d, (ii) depth K[x]/in_<rev (I _G ) = f and K[x]/in_<lex (I _G ) is Cohen–Macaulay, where in_<rev (I _G ) (resp., in_<lex (I _G )) is the initial ideal of I _G with respect to <_rev (resp., <_lex) and where depth K[x]/in_<rev (I _G ) is the depth of K[x]/in_<rev (I _G ).     

The Rees Valuations of Complete Ideals in a Regular Local Ring

Let I be a complete m-primary ideal of a regular local ring (R, m) of dimension d ≥ 2. In the case of dimension two, the beautiful theory developed by Zariski implies that I factors uniquely as a product of powers of simple complete ideals and each of the simple complete factors of I has a unique Rees valuation. In the higher dimensional case, a simple complete ideal of R often has more than one Rees valuation, and a complete m-primary ideal I may have finitely many or infinitely many base points. For the ideals having finitely many base points Lipman proves a unique factorization involving special *-simple complete ideals and possibly negative exponents of the factors. Let T be an infinitely near point to R with dim R = dim T and R/m = T/m _T . We prove that the special *-simple complete ideal P _RT has a unique Rees valuation if and only if either dim R = 2 or there is no change of direction in the unique finite sequence of local quadratic transformations from R to T. We also examine conditions for a complete ideal to be projectively full.     

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.     

Localization and Catenarity in Iterated Differential Operator Rings

ABSTRACT

Let k be a field, R an associative k-algebra with identity, Δ a finite set of derivations of R, and R[Θ₁, δ₁] ··· [Θ _n , δ _n ] an iterated differential operator k-algebra over R such that δ _j (Θ _i ) ∈ R[Θ₁, δ₁] ··· [Θ _i?1, δ _i?1]; 1 ≤ i < j ≤ n. If R is Noetherian Δ-hypercentral, then every prime ideal P of A is classically localizable. The aim of this article is to show that under some additional hypotheses on the Δ-prime ideals of R, the local ring A _P is regular in the sense of Robert Walker. We use this result to study the catenarity of A and to compute the numbers μ _i of Bass. Let g be a nilpotent Lie algebra of finite dimension n acting on R by derivations and U(g) the enveloping algebra of g. Then the crossed product of R by U(g) is an iterated differential operator k-algebra as above. In this particular case, our results are known if k has characteristic zero.     

S-Noetherian Rings of the Forms 𝒜[X] and 𝒜[[X]]

Let 𝒜 = (A _n ) _n≥0 be an ascending chain of commutative rings with identity, S ? A ₀ a multiplicative set of A ₀, and let 𝒜[X] (respectively, 𝒜[[X]]) be the ring of polynomials (respectively, power series) with coefficient of degree i in A _i for each i ∈ ?. In this paper, we give necessary and sufficient conditions for the rings 𝒜[X] and 𝒜[[X]] to be S ? Noetherian.     

Geometries and amalgams of J1

ABSTRACT

Let S = 𝕂[x ₁, …, x _n ] be a polynomial ring over a field 𝕂 and I a monomial ideal of S. It is well known that the Poincaré series of 𝕂 over S/I is rational. We describe the coefficients of the denominator of the series and study the multigraded homotopy Lie algebra of S/I.     

APPLICATIONS OF CLIFFORD ALGEBRAS TO INVOLUTIONS AND QUADRATIC FORMS

ABSTRACT

Let k be a field, char k ≠ 2, F = k(x), D a biquaternion division algebra over k, and σ an orthogonal involution on D with nontrivial discriminant. We show that there exists a quadratic form ? ∈ I ²(F) such that dim ? = 8, [C(?)] = [D], and ? does not decompose into a direct sum of two forms similar to two-fold Pfister forms. This implies in particular that the field extension F(D)/F is not excellent. Also we prove that if A is a central simple K-algebra of degree 8 with an orthogonal involution σ, then σ is hyperbolic if and only if σ _K(A) is hyperbolic. Finally, let σ be a decomposable orthogonal involution on the algebra M _{2 ^m} (K). In the case m ≤ 5 we give another proof of the fact that σ is a Pfister involution. If m ≥ 2 ^n?2 ? 2 and n ≥ 5, we show that q _σ ∈ I ⁿ (K), where q _σ is a quadratic form corresponding to σ. The last statement is founded on a deep result of Orlov et al. (2000) concerning generic splittings of quadratic forms.     

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.     

Chains of Rings with Local Formal Fibers

Let (T, M) be a complete local domain containing the integers. Let p ₁ ? p ₂ ? ··· ? p _n be a chain of nonmaximal prime ideals of T such that T _{p n} is a regular local ring. We construct a chain of excellent local domains A _n  ? A _n?1 ? ··· ? A ₁ such that for each 1 ≤ i ≤ n, the completion of A _i is T, the generic formal fiber of A _i is local with maximal ideal p _i , and if I is a nonzero ideal of A _i then A _i /I is complete. We then show that if Q is a nonmaximal prime ideal of T and 1 ≤ h = ht _T Q, then there is a chain of excellent local domains B ₀ ? B ₁ ? ··· ? B _h  ? T such that for every i = 0, 1, 2,…, h we have ht(Q ∩ B _i ) = i, the completion of B _i is isomorphic to T[[X ₁, X ₂,…, X _i ]] where the X _j 's are indeterminants, and the formal fiber of Q ∩ B _i is local.     

On the Automorphism Group of the First Weyl Algebra

Let A ₁: = 𝕜[t, ?] be the first algebra over a field 𝕜 of characteristic zero. Let Aut_𝕜(A ₁) be the automorphism group of the ring A ₁. One can associate to each right ideal I of A ₁ a subgroup of Aut_𝕜(A ₁) called the isomorphism subgroup of I. In this article, we show that each such isomorphism subgroup is equal to its normalizer. For that, we study when the isomorphism subgroup of a right ideal of A ₁ contains a given isomorphism subgroup.     

Sections of zero dimensional ideals over a Noetherian ring

Let A be a commutative Noetherian ring and be an ideal containing a monic polynomial such that A[T]/I is zero dimensional. Suppose the conormal module I/I ² is generated by r elements over A[T]/I. Then a set of r generators of can be lifted to a r generating set of I. A part of this work is done at the Abdus Salam, International Centre for Theoretical Physics, Trieste, Italy. Received: 12 March 2006 Revised: 29 January 2007     

On Monomial Reduction and Polynomial Expressions with Respect to Binomial Ideals

Let I ? K[x ₁,…, x _n ] be an ideal and G be the reduced Gröbner basis of I with respect to lexicographic monomial order. We introduce the index of an expression of f ∈ K[x ₁,…, x _n ] with respect to G. A minimal expression is characterized as the one with zero G-index. In case where I is a binomial prime ideal, a new division algorithm with minimal and unique expression is presented. The application of our new method on benchmark polynomial systems cyclic-9 and cyclic-12 shows its superiority in comparison with the existing division algorithm.     

Constructing Almost Excellent Unique Factorization Domains

Abstract

Let (T, M) be a complete local normal integral domain containing the rationals such that |T/M | ≥ c where c is the cardinality of the real numbers. Let p be a non-maximal prime ideal of T such that _{T p} is a regular local ring. We construct a local Unique Factorization Domain (UFD) A such that the M-adic completion of A is T, p is maximal in the generic formal fiber and all fibers of A are geometrically regular except for those over some height one prime ideals.

     

Characterizations of *-Cancellation Ideals of an Integral Domain

Let D be an integral domain and * a star-operation on D. For a nonzero ideal I of D, let I ^{* _f} = ?{J* | (0) ≠ J ? I is finitely generated} and I ^{* _w} = ? _{P∈* f -Max(D)} ID _P . A nonzero ideal I of D is called a *-cancellation ideal if (IA)* = (IB)* for nonzero ideals A and B of D implies A* =B*. Let X be an indeterminate over D and N _* = {f ∈ D[X] | (c(f))* =D}. We show that I is a * _w -cancellation ideal if and only if I is * _f -locally principal, if and only if ID[X] _{N *} is a cancellation ideal. As a corollary, we have that each nonzero ideal of D is a * _w -cancellation ideal if and only if D _P is a principal ideal domain for all P ∈ * _f -Max(D), if and only if D[X] _{N *} is an almost Dedekind domain. We also show that if I is a * _w -cancellation ideal of D, then I ^{* _w}  = I ^{* _f}  = I _t , and I is * _w -invertible if and only if I ^{* _w}  = J _v for a nonzero finitely generated ideal J of D.     

Minimal Generating Sets for Relative Ideals in Numerical Semigroups of Multiplicity Eight

Abstract

Let S be a numerical semigroup and let I be a relative ideal of S. Let S ? I denote the dual of I and let μ _S (?) represent the size of a minimal generating set. We investigate the inequality μ _S (I)μ _S (S ? I) ≥ μ _S (I + (S ? I)) under the assumption that S has multiplicity 8. We will show that if I is non-principal, then the strict inequality μ _S (I)μ _S (S ? I) > μ _S (I + (S ? I)) always holds.     

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.     

PrÜfer-Like Domains and the Nagata Ring of Integral Domains

A subring A of a Prüfer domain B is a globalized pseudo-valuation domain (GPVD) if (i) A?B is a unibranched extension and (ii) there exists a nonzero radical ideal I, common to A and B such that each prime ideal of A (resp., B) containing I is maximal in A (resp., B). Let D be an integral domain, X be an indeterminate over D, c(f) be the ideal of D generated by the coefficients of a polynomial f ∈ D[X], N = {f ∈ D[X] | c(f) = D}, and N _v  = {f ∈ D[X] | c(f)^?1 = D}. In this article, we study when the Nagata ring D[X] _N (more generally, D[X] _{N v} ) is a GPVD. To do this, we first use the so-called t-operation to introduce the notion of t-globalized pseudo-valuation domains (t-GPVDs). We then prove that D[X] _{N v} is a GPVD if and only if D is a t-GPVD and D[X] _{N v} has Prüfer integral closure, if and only if D[X] is a t-GPVD, if and only if each overring of D[X] _{N v} is a GPVD. As a corollary, we have that D[X] _N is a GPVD if and only if D is a GPVD and D has Prüfer integral closure. We also give several examples of integral domains D such that D[X] _{N v} is a GPVD.     

On the Factorial Closure of Certain Monomial Modules

Abstract

Let R = K[y ₁,…,y _t ] be an affine domain over a field K and I be a nonzero proper ideal of R. In Sec. 1 of this note, we characterize when (K + I, R) is a Mori pair. In Sec. 2 of this note, we prove the following theorem: Let A ? B be domains such that C/Q is Mori for each subring C of B containing A and for any prime ideal Q of C. Then dim A ? 1 ≤ dim B ≤ dim A + 1 and if dim A > 1 or dim B > 1 then dim A = dim B.