Related papers: Equimultiple Coefficient Ideals

Let $G$ be a group, $R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set of all graded ideals of $R$. Suppose that $\phi:GI(R)\rightarrow GI(R)\cup\{\emptyset\}$ is a function. In this article, we introduce and…

Let $R$ be a commutative ring with identity, $S$ a multiplicative subset of $R$ and $I$ an ideal of $R$ disjoint from $S$. In this paper, we introduce the notion of an $S$-$n$-absorbing ideal which is a generalization of both the $S$-prime…

The ring of periodic distributions on ${\mathbb{R}}^{\tt d}$ with usual addition and with convolution is considered. Via Fourier series expansions, this ring is isomorphic to the ring ${\mathcal{S}}'({\mathbb{Z}}^{\tt d})$ of all maps…

A famous result due to I. M. Isaacs states that if a commutative ring $R$ has the property that every prime ideal is principal, then every ideal of $R$ is principal. This motivates ring theorists to study commutative rings for which every…

Let $\mathcal{O}_K$ be the ring of integers of an algebraic number field $K$ embedded into $\mathbb{C}$. Let $X$ be a subset of the Euclidean space $\mathbb{R}^d$, and $D(X)$ be the set of the squared distances of two distinct points in…

Let $I$ be a monomial ideal in a polynomial ring $S=K[x_1,\ldots,x_n]$ over a field $K$ with $n=2$ or $3$, and let $\overline{I}$ be its integral closure. We will show that $\text{reg} (\overline{I}) \le \text{reg} (I)$. Furthermore, if $I$…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gr\"obner basis of the defining ideal $J$ of $A$ we give a condition, called the x-condition, which implies that all graded…

We characterise ideals in two-dimensional regular local rings that arise as ideals of maximal minors of indecomposable integrally closed modules of rank three.

In this paper, we view the collection of ideals of a commutative principal ideal ring from two perspectives: one as an ordered semigroup I(R) and the other as a category I_R . It is shown that I(R) is a regular ordered semigroup whereas I_R…

Let K be a field and let S = K[x_1, ..., x_n] be a polynomial ring. Consider a homogenous ideal I in S. Let t_i denote reg(Tor_i (S/I, K)), the maximal degree of an ith syzygy of S/I. We prove bounds on the numbers t_i for i > n/2 purely in…

Let $R$ be a $d$-dimensional standard graded ring over an Artin local ring. Let $M$ be the unique maximal homogeneous ideal of $R.$ Let $h^i(R)_n$ denote the length of $H^i_M(R)_n$, i.e. the nth graded component of the ith local cohomology…

We discuss principality of prime ideals of finite algebraic number fields $L=K(\theta)$ over an algebraic number field $K ([K:\mathbb{Q}]<\infty)$ defined by irreducible polynomials $f(x)\in \mathfrak{O}_{K}[x]$ and $f(\theta)=0$. Our main…

Let $R$ be a commutative ring, $Y\subseteq \mathrm{Spec}(R)$ and $ h_Y(S)=\{P\in Y:S\subseteq P \}$, for every $S\subseteq R$. An ideal $I$ is said to be an $\mathcal{H}_Y$-ideal whenever it follows from $h_Y(a)\subseteq h_Y(b)$ and $a\in…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field $K$, and let $A$ be a finitely generated standard graded $S$-algebra. We show that if the defining ideal of $A$ has a quadratic initial ideal, then all the graded components of…

For a submodule $N$ of an $R$-module $M$, a unique product of prime ideals in $R$ is assigned, which is called the generalized prime ideal factorization of $N$ in $M$, and denoted as ${\mathcal{P}}_M(N)$. But for a product of prime ideals…

We consider ideals $I$ in a Stanley-Reisner ring $k[\Delta]$ over the simplical complex $\Delta$, such that the tight closure of $I$, $I^*$, is equal to $\mathfrak{m}$, the standard graded maximal ideal of $k[\Delta]$. We determine the…

Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an…

Using perfectoid algebras, we introduce a mixed characteristic analog of the multiplier ideal, respectively test ideal, from characteristic zero, respectively $p > 0$, in the case of a regular ambient ring. We prove several properties about…

Let $R$ be a commutative ring with identity. For an $R$-module $M$, the notion of strongly prime submodule of $M$ is defined. It is shown that this notion of prime submodule inherits most of the essential properties of the usual notion of…

A semiring is uniserial if its ideals are totally ordered by inclusion. First, we show that a semiring $S$ is uniserial if and only if the matrix semiring $M_n(S)$ is uniserial. As a generalization of valuation semirings, we also…