交换代数

Let $(A,\mathfrak{m})$ be a hypersurface local ring of dimension $d \geq 1$, $N$ a perfect $A$-module and let $I$ be an ideal in $A$ with $\ell(N/IN)$ finite. We show that there is a integer $r_I \geq -1$ (depending only on $I$ and $N$)…

It is proved that for any finite dimensional representation of a prime order group over the field of rational numbers, polynomial invariants of degree at most $3$ separate the orbits. A result providing an upper degree bound for separating…

We study a class of double determinantal ideals denoted $I_{mn}^r$, which are generated by minors of size 2, and show that they are equal to the Hibi rings of certain finite distributive lattices. We compute the number of minimal generators…

Let $R$ be a commutative ring with. The purpose of this paper is to introduce and investigate cubes-difference factor absorbing ideals of R as a generalization of prime ideals.

Let $R$ be a Noetherian local ring of Krull dimension $d$ such that $(d!)R = R$, and let $A$ be a graded $R$-subalgebra of the polynomial algebra $R[t]$. We prove that every unimodular row of length $d + 1$ over $A$ can be completed to an…

Let $x_1,\ldots,x_s$ be a filter regular sequence in a local ring $(R,\mathfrak{m})$. Denote by $R_{x_1,\ldots,x_s}$ the Koszul complex of $x_1,\ldots,x_s$ over $R$. In this paper, we give an explicit number $N$ such that the sum of lengths…

Let $R$ be a local ring with residue field $k$ and $M$, $N$ be finitely generated modules over $R$. It is well known that $Tor^R_i(M, N) = 0$ for $i \gg 0$ if $pd_R(M) < \infty$ or $pd_R(N) < \infty$. The ring $R$ is said to satisfy the…

It is well-known that within Zermelo-Fraenkel set theory (ZF), the Axiom of Choice (AC) implies the Maximal Ideal Theorem (MIT), namely that every nontrivial commutative ring has a maximal ideal. The converse implication MIT $\Rightarrow$…

Let $R=\k[x,y,z]$ and $I=(f_0,\dots,f_{n-1})$ be a height two perfect ideal which is almost linearly presented (that is, all but the last column have linear entries, but the last column has entries which are homogeneous of degree $2$).…

Projective monomial curves are associated with rings generated by monomials of equal degree in two variables. In this paper, we give an infinite class of non-smooth, non Cohen-Macaulay $k$-Buchsbaum projective monomial curves for any $k\geq…

This is the second in a series of papers highlighting the applications of reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ be an ideal of $R$. We give necessary and sufficient conditions in terms of $I$-reduced…

We discuss the relationship between the trace ideal of the canonical module and pseudo-Gorensteinness. In particular, under certain mild assumptions, we show that every pseudo-Gorenstein nearly Gorenstein graded domain is Gorenstein. As an…

Let $R$ be a Noetherian ring of dimension $d$ and $A$ be a graded $R$-subalgebra of $R[X,1/X]$. Let $P$ be a projective module over $A$ of rank $r \geq \max\{d+1,2\}$ and $\v=(a,p)$ be a unimodular element of $A \oplus P$. We find an…

In this paper, we define a new concept of Noetherian commutative rings which stands between Gorenstein and Cohen-Macaulay properties. We show that this new property keep hold under common operations of commutative rings such as…

We introduce the notions of sequential sequence and sequential f-sequence in order to characterize sequentially Cohen-Macaulay modules and sequentially generalized Cohen-Macaulay modules. Let R be a Noetherian local ring and M a finitely…

Let $\mathbb{M} = \{ M_n \}$ be a good $\mathfrak{q}$-filtration of a finitely generated $R$-module $M$ of dimension $d$, where $(R,\mathfrak{m})$ is a local ring and $\mathfrak{q}$ is an $\mathfrak{m}$-primary ideal of $R$. In case…

Let $C,A$ be countable abelian groups. In this paper we determine the complexity of classifying extensions $C$ by $A$, in the cases when $C$ is torsion-free and $A$ is a $p$-group, a torsion group with bounded primary components, or a free…

We introduce an algebraic model, based on the determinantal expansion of the product of two matrices, to test combinatorial reductions of set functions. Each term of the determinantal expansion is deformed through a monomial factor in d…

Let $K$ be an infinite field and let $I = (f_1,\cdots,f_r)$ be an ideal in the polynomial ring $R = K[x_1,\cdots,x_n]$ generated by generic forms of degrees $d_1,\cdots,d_r$. A longstanding conjecture by Fr\"{o}berg predicts the shape of…

An $\mathcal{A}$-semigroup is a numerical semigroup without consecutive small elements. This work generalizes this concept to finite-complement submonoids of an affine cone $\mathcal{C}$. We develop algorithmic procedures to compute all…