On the S 2-Fication of Some Toric Varieties

In this article we prove the following:

1. Some results on the Cohen–Macaulayness of the canonical module;

2. We study the S ₂-fication of rings which are quotients by lattices ideals;

3. Given a simplicial lattice ideal of codimension two I, its Macaulayfication is given explicitly from a system of generators of I.

     

Quasi-Schreier Domains II

We study a class of integral domains characterized by the property that every nonzero finite intersection of principal ideals is a directed union of invertible ideals.     

Rees Matrix Covers and the Translational Hull of a Locally Inverse Semigroup

For any locally inverse semigroup S, there exists a maximal dense ideal extension of S within the class LI of all locally inverse semigroups (Pastijn and Oliveira, 2006 Pastijn , F. J. , Oliveira , L. ( 2006 ). Maximal dense ideal extensions of locally inverse semigroups . Semigroup Forum 72 : 441 – 458 . [Google Scholar], Preprint). Here we realize this maximal dense ideal extension in terms of a canonically constructed quotient of a regular Rees matrix semigroup over an inverse semigroup.     

On Canonical Ideals of One-Dimensional Local Analytically Irreducible Rings

A characterization is given for certain canonical ideals of local one-dimensional commutative rings with identity under the assumption of analytical irreducibility. This extends both a known result characterizing the case when the ring itself is a canonical ideal (i.e., it is a Gorenstein ring), and another one which is a general characterization of canonical ideals but under the further assumption of residual rationality.     

½ Cancellation Modules and Homogeneous Idealization II

In our recent work we investigated ½ (weak) cancellation modules and ½ join principal submodules and showed via the method of idealization most questions concerning these modules can be reduced to the ideal case. The purpose of this article is to continue our study of these modules as well as we introduce and give some properties of the concept of M-join principal ideals.     

Unitary Numerical Semigroups and Perfect Bricks

This article completes a previous investigation of balanced and unitary numerical semigroups. The main result establishes the equivalence of unitary numerical semigroups and perfect 2 × 2 bricks.     

On Generalized Stable Ideals

An ideal I of a ring R is generalized stable in case aR + bR = R with a ∈ I, b ∈ R implies that there exist s, t ∈ 1 + I such that s(a + by)t = 1 for a y ∈ R. We establish, in this article, necessary and sufficient conditions for an ideal of a regular ring to be generalized stable. It is shown that every regular square matrix over such ideals admits a diagonal reduction. These extend the corresponding results of generalized stable regular rings.     

Depths of Powers of the Edge Ideal of a Tree

Lower bounds are given for the depths of R/I ^t for t ≥ 1 when I is the edge ideal of a tree or forest. The bounds are given in terms of the diameter of the tree, or in case of a forest, the largest diameter of a connected component and the number of connected components. These lower bounds provide a lower bound on the power for which the depths stabilize.     

Primes and Right Ideals in Right Cones

A right cone H in a group G is a submonoid of G that generates G and aH ? bH for a, b ? H implies bH ? aH. With any right ideal I ≠ H of H a completely prime ideal P _r (I) of H is associated and the set 𝒫(I) of right ideals I′ of H with the same associated prime ideal P′ =P _r (I) is determined if P′·? P″ is a right invariant segment in H. The set 𝒫(I) is also described if P _r (I) is a limit prime.     

Nested Log-Concavity

We give conditions on the coefficients of a polynomial p(x) so that p(x + t) be log-concave or strictly log-concave. Several applications are given: if p(x) is a polynomial with nonnegative and nondecreasing coefficients, then p(x + t) is strictly log-concave for all t ≥ 1; for any polynomial p(x) with positive leading coefficient, there is t ₀ ≥ 0 such that for any t ≥ t ₀ it holds that the coefficients of p(x + t) are positive, strictly decreasing, and strictly log-concave; if p(x) is a log-concave polynomial with nonnegative coefficients and no internal zeros, then p(x + t) is strictly log-concave for all t > 0; Betti numbers of lexsegment monomial ideals are strictly log-concave.