Stanley-Reisner rings and the radicals of lattice ideals

In this article we associate to every lattice ideal I_L,ρ⊂K[x₁,…,x_m] a cone σ and a simplicial complex Δ_σ with vertices the minimal generators of the Stanley-Reisner ideal of σ. We assign a simplicial subcomplex Δ_σ(F) of Δ_σ to every polynomial F. If F₁,…,F_s generate I_L,ρ or they generate rad(I_L,ρ) up to radical, then is a spanning subcomplex of Δ_σ. This result provides a lower bound for the minimal number of generators of I_L,ρ which improves the generalized Krull's principal ideal theorem for lattice ideals. But mainly it provides lower bounds for the binomial arithmetical rank and the A-homogeneous arithmetical rank of a lattice ideal. Finally, we show by a family of examples that the given bounds are sharp.     

Resolutions of Stanley-Reisner rings and Alexander duality

Associated to any simplicial complex Δ on n vertices is a square-free monomial ideal I_Δ in the polynomial ring A = k[x₁, …, x_n], and its quotient k[Δ] = A/I_Δ known as the Stanley-Reisner ring. This note considers a simplicial complex Δ^* which is in a sense a canonical Alexander dual to Δ, previously considered in [1, 5]. Using Alexander duality and a result of Hochster computing the Betti numbers dim_k Tor_i^A (k[Δ],k), it is shown (Proposition 1) that these Betti numbers are computable from the homology of links of faces in Δ^*. As corollaries, we prove that I_Δ has a linear resolution as A-module if and only if Δ^* is Cohen-Macaulay over k, and show how to compute the Betti numbers dim_k Tor_i^A (k[Δ],k) in some cases where Δ^* is wellbehaved (shellable, Cohen-Macaulay, or Buchsbaum). Some other applications of the notion of shellability are also discussed.     

Neat rings

A ring is called clean if every element is the sum of a unit and an idempotent. Throughout the last 30 years several characterizations of commutative clean rings have been given. We have compiled a thorough list, including some new equivalences, in hopes that in the future there will be a better understanding of this interesting class of rings. One of the fundamental properties of clean rings is that every homomorphic image of a clean ring is clean. We define a neat ring to be one for which every proper homomorphic image is clean. In particular, the ring of integers, Z, and any nonlocal PID are examples neat rings which are not clean. We characterize neat Bézout domains using the group of divisibility. In particular, it is shown that a neat Bézout domain has stranded primes, that is, for every nonzero prime ideal the set of primes either containing or contained in the given prime forms a chain under set-theoretic inclusion.     

Strong Mori and Noetherian properties of integer-valued polynomial rings

Let D be a domain with quotient field K and let Int(D) be the ring of integer-valued polynomials {f∈K[X]|f(D)⊆D}. We give conditions on D so that the ring Int(D) is a Strong Mori domain. In particular, we give a complete characterization in the case that the conductor is nonzero, where D′ is the integral closure of D. We also show that when D is quasilocal with or D is Noetherian, Int(D) is a Strong Mori domain if and only if Int(D) is Noetherian.     

On the class group and S-class group of formal power series rings

Let ${\text{Cl}}_t(A)$ denote the t-class group of an integral domain A. P. Samuel has established that if A is a Krull domain then the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is injective and if A is a regular UFD, then ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is bijective. Later, L. Claborn extended this result in case A is a regular Noetherian domain. In the first part of this paper we prove that the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$; $[I]?[{(I.A?X?)}_t]$ is an injective homomorphism and in case of an integral domain A such that each υ-invertible υ-ideal of A has υ-finite type, we give an equivalent condition for ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, to be bijective, thus generalizing the result of Claborn. In the second part of this paper, we define the S-class group of an integral domain A: let S be a (not necessarily saturated) multiplicative subset of an integral domain A. Following [11], a nonzero fractional ideal I of A is S-principal if there exist an $s\in S$ and $a\in I$ such that $sI?aA?I$. The S-class group of A, S-${\text{Cl}}_t(A)$ is the group of fractional t-invertible t-ideals of A under t-multiplication modulo its subgroup of S-principal t-invertible t-ideals of A. We generalize some known results developed for the classic contexts of Krull and PυMD domain and we investigate the case of isomorphism S-${\text{Cl}}_t(A)?S$-${\text{Cl}}_t(A?X?)$.     

Bézout rings with almost stable range 1

Elementary divisor domains were defined by Kaplansky [I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949) 464-491] and generalized to rings with zero-divisors by Gillman and Henriksen [L. Gillman, M. Henriksen, Some remarks about elementary divisor rings, Trans. Amer. Math. Soc. 82 (1956) 362-365]. In [M.D. Larsen, W.J. Lewis, T.S. Shores, Elementary divisor rings and finitely presented modules, Trans. Amer. Math. Soc. 187 (1) (1974) 231-248], it was also proved that if a Hermite ring satisfies (N), then it is an elementary divisor ring. The aim of this article is to generalize this result (as well as others) to a much wider class of rings. Our main result is that Bézout rings whose proper homomorphic images all have stable range 1 (in particular, neat rings) are elementary divisor rings.     

Full ideals of polynomial rings

An idealI of the ringK[x ₁, ...,x _n ] of polynomials over a fieldK inn indeterminates is a full ideal ifI is closed under substitution,f I,g ₁...g_n K[x ₁, ...,x _n ] implyf(g ₁, ...,g _n ) I. In this paper we continue the investigation of full ideals ofK[x ₁, ...,x _n ]. In particular we determine several classes of full ideals ofK[x, y] (K a finite field) and investigate properties of these classes.The first author gratefully acknowledges support from theDeutsche Forschungsgemeinschaft     

Almost slender rings and compact rings

Classical results concerning slenderness for commutative integral domains are generalized to commutative rings with zero divisors. This is done by extending the methods from the domain case and bringing them in connection with results on the linear topologies associated to non-discrete Hausdorff filtrations. In many cases a weakened notion “almost slenderness” of slenderness is appropriate for rings with zero divisors. Special results for countable rings are extended to rings said to be of “bounded type” (including countable rings, ‘small’ rings, and, for instance, rings that are countably generated as algebras over an Artinian ring).More precisely, for a ring R of bounded type it is proved that R is slender if R is reduced and has no simple ideals, or if R is Noetherian and has no simple ideals; moreover, R is almost slender if R is not perfect (in the sense of H. Bass). We use our methods to study various special classes of rings, for instance von Neumann regular rings and valuation rings. Among other results we show that the following two rings are slender: the ring of Puiseux series over a field and the von Neumann regular ring $k^{\mathbb{N}}/k^{(\mathbb{N})}$ over a von Neumann regular ring k.For a Noetherian ring R we prove that R is a finite product of local complete rings iff R satisfies one of several (equivalent) conditions of algebraic compactness. A 1-dimensional Noetherian ring is outside this ‘compact’ class precisely when it is almost slender. For the rings of classical algebraic geometry we prove that a localization of an algebra finitely generated over a field is either Artinian or almost slender. Finally, we show that a Noetherian ring R is a finite product of local complete rings with finite residue fields exactly when there exists a map of R-algebras $R^{\mathbb{N}}\to R$ vanishing on $R^{(\mathbb{N})}$.     

Combinatorial properties of virtual braids

We determine the lower central series of the virtual braid group VBn and of the kernels of two different projections of VBn in Sn: the normal closure of the Artin braid group Bn, that we will denote by Hn, and the so-called virtual pure braid group VPn, which is related to Yang Baxter equation. We describe relations between Hn and VPn and we provide a connection between virtual pure braids and the finite type invariant theory for virtual knots defined by Goussarov, Polyak and Viro.     

Noetherian-like properties in polynomial and power series rings

There are many Noetherian-like rings. Among them, we are interested in SFT-rings, piecewise Noetherian rings, and rings with Noetherian prime spectrum. Some of them are stable under polynomial extensions but none of them are stable under power series extensions. We give partial answers to some open questions related with stabilities of such rings. In particular, we show that any mixed extensions $R[X_1??[X_n?$ over a zero-dimensional SFT ring R are also SFT-rings, and that if R is an SFT-domain such that $R/P$ is integrally closed for each prime ideal P of R, then $R[X]$ is an SFT-ring. We also give a direct proof that if R is an SFT Prüfer domain, then $R[X_1,?,X_n]$ is an SFT-ring. Finally, we show that the power series extension $R?X?$ over a Prüfer domain R is piecewise Noetherian if and only if R is Noetherian.     

An algorithm for unimodular completion over Laurent polynomial rings

We present a new and simple algorithm for completion of unimodular vectors with entries in a multivariate Laurent polynomial ring over an infinite field K. More precisely, given n?3 and a unimodular vector V=^t(v₁,…,v_n)∈Rⁿ (that is, such that 〈v₁,…,v_n〉=R), the algorithm computes a matrix M in M_n(R) whose determinant is a monomial such that MV=^t(1,0,…,0), and thus M^-1 is a completion of V to an invertible matrix.     

Cotangent cohomology of Stanley-Reisner rings

Simplicial complexes X provide commutative rings A(X) via the Stanley- Reisner construction. We calculated the cotangent cohomology, i.e., T¹ and T² of A(X) in terms of X. These modules provide information about the deformation theory of the algebro geometric objects assigned to X.     

Algebraic series and valuation rings over nonclosed fields

Suppose that k is an arbitrary field. Let k[[x₁,…,x_n]] be the ring of formal power series in n variables with coefficients in k. Let be the algebraic closure of k and . We give a simple necessary and sufficient condition for σ to be algebraic over the quotient field of k[[x₁,…,x_n]]. We also characterize valuation rings V dominating an excellent Noetherian local domain R of dimension 2, and such that the rank increases after passing to the completion of a birational extension of R. This is a generalization of the characterization given by M. Spivakovsky [Valuations in function fields of surfaces, Amer. J. Math. 112 (1990) 107–156] in the case when the residue field of R is algebraically closed.     

Differential operators on Stanley-Reisner rings

Let be an algebraically closed field of characteristic zero, and let be a polynomial ring. Suppose that is an ideal in that may be generated by monomials. We investigate the ring of differential operators on the ring , and , the idealiser of in . We show that and are always right Noetherian rings. If is a square-free monomial ideal then we also identify all the two-sided ideals of . To each simplicial complex on there is a corresponding square-free monomial ideal , and the Stanley-Reisner ring associated to is defined to be . We find necessary and sufficient conditions on for to be left Noetherian.

     

Pure-injective hulls of modules over valuation rings

If is the pure-injective hull of a valuation ring R, it is proved that is the pure-injective hull of M, for every finitely generated R-module M. Moreover , where (A_k)_1≤k≤n is the annihilator sequence of M. The pure-injective hulls of uniserial or polyserial modules are also investigated. Any two pure-composition series of a countably generated polyserial module are isomorphic.     

Algebraic properties of stably numerical polynomials

We study algebraic properties of certain rings of polynomials closely related to the ring of integer-valued polynomials. We give generators and relations for the localisations at a prime. We describe the maximal and prime ideals and show that the rings considered are not Noetherian.