交换代数

The purpose of this note is to pose a question that, when answered, would directly imply the Cohen Structure Theorem. We provide a solution to this question for a specific class of local rings (not necessarily complete). We also explore how…

We compute the regularity of powers and symbolic powers of edge ideals of all cubic circulant graphs. In particular, we establish Conjecture of Minh for cubic circulant graphs.

Let $\Gamma$ be a torsionless commutative cancellative monoid, $R=\bigoplus_{\alpha \in \Gamma}R_{\alpha}$ be a $\Gamma$-graded integral domain. In this note we show that each homogeneous star operation…

In this paper, we investigate the internal (finite) sum of submonoids of rank-$1$ torsion-free abelian groups. These submonoids, when not groups, are isomorphic to nontrivial submonoids of the nonnegative cone of $\mathbb Q$, known as…

We give sufficent conditions for a derivation of a $k$-algebra $A$ of finite type to be $\infty$-integrable in the sense of Hasse-Schmidt, when $A$ is a complete intersection, or when $A$ is reduced and $k$ is a regular ring. As a…

A commutative cancellative monoid is atomic if every non-invertible element factors into irreducibles (also called atoms), while an integral domain is atomic if its multiplicative monoid is atomic. Back in the eighties, Gilmer posed the…

In this work, we introduce the concept of relative Lipschitz saturation, along with its key categorical and algebraic properties, and demonstrate how such a structure always gives rise to a radicial algebra.

This paper extends the results of Boij, Eisenbud, Erman, Schreyer, and S\"oderberg on the structure of Betti cones of finitely generated graded modules and finite free complexes over polynomial rings, to all finitely generated graded rings…

We consider ideals in a polynomial ring generated by collections of power sum polynomials, and obtain conditions under which these define complete intersection rings, normal domains, and unique factorization domains. We also settle a key…

Let $R$ be a commutative Noetherian ring of dimension $d$. In this paper, we first show that some power of the cohomology annihilator annihilates the $(d+1)$-th Ext modules for all finitely generated modules when either $R$ admits a…

We completely determine the depth and regularity of symbolic powers of the fiber product of two homogeneous ideals in disjoint sets of variables, given knowledge of the symbolic powers of each factor. Generalizing previous joint work with…

Ideals in the ring of power series in three variables can be classified based on algebra structures on their minimal free resolutions. The classification is incomplete in the sense that it remains open which algebra structures actually…

Given a generically finite local extension of valuation rings $V \subset W$, the question of whether $W$ is the localization of a finitely generated $V$-algebra is significant for approaches to the problem of local uniformization of…

Our aim in this paper is to explore semisubtractive ideals of semirings. We prove that they form a complete modular lattice. We introduce Golan closures and prove some of their basic properties. We explore the relations between $Q$-ideals…

Let $(A,\m)$ be a Cohen-Macaulay local ring of dimension $d\geq 3$, $I$ an $\m$-primary ideal and $\mathcal{I}=\{I_n\}_{n\geq 0}$ an $I$-admissible filtration. We establish bounds for the third Hilbert coefficient: (i) $e_3(\mathcal{I})\leq…

Given two ideals $I$ and $J$ of the ring $\mathcal O_n$ of analytic function germs $f:(\mathbb C^n,0)\to \mathbb C$, we show a sharp lower bound for the log canonical threshold of $IJ$ in terms of the sequences of mixed {\L}ojasiewicz…

Let $E$ be a module of projective dimension one over $R=k[x_1,\ldots,x_d]$. If $E$ is presented by a matrix $\varphi$ with linear entries and the number of generators of $E$ is bounded locally up to codimension $d-1$, the Rees ring…

In this paper, we study the solving degrees for affine semi-regular sequences and their homogenized sequences. Some of our results are considered to give mathematically rigorous proofs of the correctness of methods for computing Gr\"{o}bner…

Let R be a commutative ring with identity, and let S be a multiplicative subset of R. Positselski and Sl\'avik introduced the concepts of S-strongly flat modules and S-weakly cotorsion R-modules, and they showed that these concepts are…

Blockwise determinantal ideals are those generated by the union of all the minors of specified sizes in certain blocks of a generic matrix, and they are the natural generalization of many existing determinantal ideals like the Schubert and…