Mathematics

The purpose of this paper is to prove that the symbolic Rees rings of ideals defining certain finite sets of points in the projective plane over an algebraically closed field are finitely generated using a ring theoretical criterion which is known as Huneke's criterion.

We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power series of bounded complexity lying on an algebraic variety defined over the field of power series.

We consider flat epimorphisms of commutative rings R\to U such that, for every ideal I\subset R for which IU=U , the quotient ring R/I is semilocal of Krull dimension zero. Under these assumptions, we show that the projective dimension of the R -module U does not exceed 1 . We also describe the Geigle-Lenzing perpendicular subcategory U^{\perp_{0,1}} in R\mathsf{-Mod} . Assuming additionally that the ring U and all the rings R/I are perfect, we show that all flat R -modules are U -strongly flat. Thus we obtain a generalization of some results of the paper arXiv:1801.04820, where the case of the localization U=S^{-1}R of the ring R at a multiplicative subset S\subset R was considered.

Let (R,m) be a Noetherian local ring of prime characteristic p>0 , and t an integer such that H j m (R)/ 0 F H j m (R) has finite length for all j<t . The aim of this paper is to show that there exists an uniform bound for Frobenius test exponents of ideals generated by filter regular sequences of length at most t .

Let R be be a commutative ring with identity and S be a multiplicatively closed subset of R. In this article we introduce the concepts of S-coidempotent submodules and fully S-coidempotent R-modules as generalizations of coidempotent submodules and fully coidempotent R-modules. We explore some basic properties of these classes of R-modules.

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The aim of this paper is to introduce the notion of fully S-idempotent modules as a generalization of fully idempotent modules and investigate some properties of this class of modules.

This paper establishes the fundamental properties of the s -closures, a recently introduced family of closure operations on ideals of rings of positive characteristic. The behavior of the s -closure of homogeneous ideals in graded rings is studied, and criteria are given for when the s -closure of an ideal can be described exactly in terms of its tight closure and rational powers. Sufficient conditions are established for the weak s -closure to equal to the s -closure. A generalization of the Briancon-Skoda theorem is given which compares any two different s -closures applied to powers of the same ideal.

We derive polynomial identities of arbitrary degree n for syzygies degrees of numerical semigroups S_m=<d_1,...,d_m> and show that for n>=m they contain higher genera G_r=\sum_{s\in Z_>\setminus S_m}s^r of S_m. We find a number g_m=B_m-m+1 of algebraically independent genera G_r and equations, related any of g_m+1 genera, where B_m=\sum_{k=1}^{m-1}\beta_k and \beta_k denote the total and partial Betti numbers of non-symmetric semigroups. The number g_m is strongly dependent on symmetry of S_m and decreases for symmetric semigroups and complete intersections.

In this article, we introduce and study the concept of ϕ -2-absorbing quasi primary ideals in commutative rings. Let R be a commutative ring with a nonzero identity and L(R) be the lattice of all ideals of R . Suppose that ϕ:L(R)→L(R)∪{∅} is a function. A proper ideal I of R is called a ϕ -2-absorbing quasiprimary ideal of R if a,b,c∈R and whenever abc∈I−ϕ(I), then either ab∈ I – √ or ac∈ I – √ or bc∈ I – √ . In addition to giving many properties of ϕ -2-absorbing quasi primary ideals, we also use them to characterize von Neumann regular rings.

We give several criteria for a ring to be a UFD including generalizations of some criteria due to P. Samuel. These criteria are applied to construct, for any field k, (1) a Z-graded non-noetherian rational UFD of dimension three over k, and (2) k-affine rational UFDs defined by trinomial relations.