共查询到20条相似文献,搜索用时 31 毫秒
1.
Oscar Fernández-Ramos 《代数通讯》2013,41(6):1921-1933
Consider an ideal I ? K[x 1,…, x n ], with K an arbitrary field, generated by monomials of degree two. Assuming that I does not have a linear resolution, we determine the step s of the minimal graded free resolution of I where nonlinear syzygies first appear, we show that at this step of the resolution nonlinear syzygies are concentrated in degree s + 3, and we compute the corresponding graded Betti number β s, s+3. The multidegrees of these nonlinear syzygies are also determined and the corresponding multigraded Betti numbers are shown to be all equal to 1. 相似文献
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Reinhold Hübl 《代数通讯》2013,41(10):3771-3781
All monomial ideals I ? k[X 0,…, X d ] are classified which satisfy the following condition: If f ∈ I with f n ∈ I n+1 for some n, then f ∈ (X 0,…, X d ) I. 相似文献
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A. R. Nasr-Isfahani 《代数通讯》2013,41(11):4461-4469
For a ring R, endomorphism α of R and positive integer n we define a skew triangular matrix ring T n (R, α). By using an ideal theory of a skew triangular matrix ring T n (R, α) we can determine prime, primitive, maximal ideals and radicals of the ring R[x; α]/ ? x n ?, for each positive integer n, where R[x; α] is the skew polynomial ring, and ? x n ? is the ideal generated by x n . 相似文献
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Let I be a homogeneous ideal of a polynomial ring K[x1,…, xn] 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(Im) ≤ 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. 相似文献
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Sarah Wolff 《代数通讯》2013,41(5):2114-2125
We specify a class of graphs, H t , and characterize the irreducible decompositions of all powers of the cover ideals. This gives insight into the structure and stabilization of the corresponding associated primes; specifically, providing an answer to the question “For each integer t ≥ 0, does there exist a (hyper) graph H t such that stabilization of associated primes occurs at n ≥ (χ(H t ) ?1) + t?” [4]. For each t, H t has chromatic number 3 and associated primes that stabilize at n = 2 + t. 相似文献
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George F. Seelinger 《代数通讯》2013,41(1):237-248
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John C. Harris 《代数通讯》2013,41(11):4278-4289
Let G be the cyclic group of order n, and suppose F is a field containing a primitive nth root of unity. We consider the ring of invariants F[W] G of a three dimensional representation W of G where G ? SL(W). We describe minimal generators and relations for this ring and prove that the lead terms of the relations are quadratic. These minimal generators for the relations form a Gröbner basis with a surprisingly simple combinatorial structure. We describe the graded Betti numbers for a minimal free resolution of F[W] G . The case where W is any two dimensional representation of G is also handled. 相似文献
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《代数通讯》2013,41(12):6115-6134
Abstract We give some techniques to determine the ideal K I generated by the monomials x k 1 y k 2 belonging to the integral closure ī of an ideal I ? ?{x, y}. We also give a sufficient condition for a weighted homogeneous ideal I ? ?{x, y} to satisfy the relation ī = I + K I . 相似文献
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Dariush Kiani 《代数通讯》2013,41(12):5376-5394
Let R = k[x1,…, xn], where k is a field. The path ideal (of length t) of a directed graph G is the monomial ideal, denoted by It(G), whose generators correspond to the directed paths of length t in G. We determine all the graded Betti numbers of the path ideal of a directed rooted tree with respect to some graphical terms. 相似文献
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We denote by 𝒜(R) the class of all Artinian R-modules and by 𝒩(R) the class of all Noetherian R-modules. It is shown that 𝒜(R) ? 𝒩(R) (𝒩(R) ? 𝒜(R)) if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)), for all centrally prime ideals P (i.e., ab ∈ P, a or b in the center of R, then a ∈ P or b ∈ P). Equivalently, if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)) for all normal prime ideals P of R (i.e., ab ∈ P, a, b normalize R, then a ∈ P or b ∈ P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, 𝒜(R) ? 𝒩(R) implies that 𝒩(R) = 𝒜(R), where R is a duo ring. For a ring R, we prove that 𝒩(R) = 𝒜(R) if and only if the coincidence in the title occurs. Finally, if Q is the quotient field of a discrete valuation domain R, it is shown that Q is the only R-module which is both α-atomic and β-critical for some ordinals α,β ≥ 1 and in fact α = β = 1. 相似文献
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Let R be a commutative ring with identity, Z(R) its set of zero-divisors, and Nil(R) its ideal of nilpotent elements. The zero-divisor graph of R is Γ(R) = Z(R)\{0}, with distinct vertices x and y adjacent if and only if xy = 0. In this article, we study Γ(R) for rings R with nonzero zero-divisors which satisfy certain divisibility conditions between elements of R or comparability conditions between ideals or prime ideals of R. These rings include chained rings, rings R whose prime ideals contained in Z(R) are linearly ordered, and rings R such that {0} ≠ Nil(R) ? zR for all z ∈ Z(R)\Nil(R). 相似文献
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《代数通讯》2013,41(3):837-854
ABSTRACT Let 𝕂 be a (commutative) field and consider a nonzero element q in 𝕂 that is not a root of unity. Goodearl and Lenagan (2002) have shown that the number of ?-primes in R = O q (? n (𝕂)) that contain all (t + 1) × (t + 1) quantum minors but not all t × t quantum minors is a perfect square. The aim of this paper is to make precise their result: we prove that this number is equal to (t!) 2 S(n + 1, t + 1)2, where S(n + 1, t + 1) denotes the Stirling number of the second kind associated to n + 1 and t + 1. This result was conjectured by Goodearl, Lenagan, and McCammond. The proof involves some closed formulas for the poly-Bernoulli numbers that were established by Kaneko (1997) and Arakawa and Kaneko (1999). 相似文献
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In this article, we study specializations of multigradings and apply them to the problem of the computation of the arithmetical rank of a lattice ideal I L 𝒢 ? K[x 1,…, x n ]. The arithmetical rank of I L 𝒢 equals the ?-homogeneous arithmetical rank of I L 𝒢 , for an appropriate specialization ? of 𝒢. To the lattice ideal I L 𝒢 and every specialization ? of 𝒢, we associate a simplicial complex. We prove that combinatorial invariants of the simplicial complex provide lower bounds for the ?-homogeneous arithmetical rank of I L 𝒢 . 相似文献
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Ignacio Ojeda Martínez de Castilla 《代数通讯》2013,41(1):279-287
A commutative ring R is said to be strongly Hopfian if the chain of annihilators ann(a) ? ann(a 2) ? … stabilizes for each a ∈ R. In this article, we are interested in the class of strongly Hopfian rings and the transfer of this property from a commutative ring R to the ring of the power series R[[X]]. We provide an example of a strongly Hopfian ring R such that R[[X]] is not strongly Hopfian. We give some necessary and sufficient conditions for R[[X]] to be strongly Hopfian. 相似文献
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We say a ring R is (centrally) generalized left annihilator of principal ideal is pure (APP) if the left annihilator ? R (Ra) n is (centrally) right s-unital for every element a ∈ R and some positive integer n. The class of generalized left APP-rings includes generalized left (principally) quasi-Baer rings and left APP-rings (and hence left p.q.-Baer rings, right p.q.-Baer rings, and right PP-rings). The class of centrally generalized left APP-rings is closed under finite direct products, full matrix rings, and Morita invariance. The behavior of the (centrally) generalized left APP condition is investigated with respect to various constructions and extensions, and it is used to generalize many results on generalized PP-rings with IFP and semiprime left APP-rings. Moreover, we extend a theorem of Kist for commutative PP rings to centrally generalized left APP rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. Furthermore, we give a complete characterization of a considerably large family of centrally generalized left APP rings which have a sheaf representation. 相似文献
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