Related papers: $F$-singularities under generic linkage

Let I and J be homogeneous ideals in a standard graded polynomial ring. We study upper bounds of the Hilbert function of the intersection of I and g(J), where g is a general change of coordinates. Our main result gives a generalization of…

Let f_1,...,f_r be homogeneous polynomials in K[x_1,...,x_n], K a field. Put F=y_1f_1+...+y_rf_r in K[x,y] and let I be the ideal of K[x,y] generated by the partials of F relative to the x_i and y_j. The Jacobian ring of F is the quotient…

Given a singular modulus $j_0$ and a set of rational primes $S$, we study the problem of effectively determining the set of singular moduli $j$ such that $j-j_0$ is an $S$-unit. For every $j_0 \neq 0$, we provide an effective way of finding…

Let $K$ be a field and $I$ a monomial ideal of the polynomial ring $S=K[x_1,..., x_n]$ generated by monomials $u_1,u_2,..., u_t$. We show that $S/I$ is pretty clean if either: 1) $u_1,u_2,..., u_t$ is a filter-regular sequence, 2)…

A polynomial with coefficients in the ring of integers $\mathcal{O}_{K}$ of a global field $K$ is called intersective if it has a root modulo every finite-indexed subgroup of $\mathcal{O}_{K}$. We prove two criteria for a polynomial…

Let $R=\oplus_{m\geq 0}R_m$ be a standard graded equidimensional ring over a field $R_0$, and $I\subseteq J$ be two non-nilpotent graded ideals in $R$. Then we give a set of numerical characterizations of the integral dependence of $I$ and…

Let R=k[x_1,...,x_n] be a polynomial ring over a field k. Let J={j_1,...,j_t} be a subset of [n]={1,...,n}, and let m_J denote the ideal (x_{j_1},...,x_{j_t}) of R. Given subsets J_1,...,J_s of [n] and positive integers a_1,...,a_s, we…

Let $R$ be a polynomial ring over a field of characteristic zero and let $I \subset R$ be a graded ideal of height $N$ which is minimally generated by $N+1$ homogeneous polynomials. If $I=(f_1,...,f_{N+1})$ where $f_i$ has degree $d_i$ and…

For a commutative unital ring $R$ with fixed ideals $I$ and $J$, we introduce and study $I$-prime $R$-modules and $(I, J)$-prime $R$-modules together with their duals $I$-coprime $R$-modules and $(I,J)$-coprime $R$-modules respectively. We…

Let $(R,\frak m)$ be an excellent generalized Cohen-Macaulay local ring of dimension $d$ that is $F$-injective on the punctured spectrum. Let $\frak q$ be a standard parameter ideal of $R$. The aim of the paper is to prove that…

Let $A$ be a regular ring of dimension $d$ essentially of finite type over an infinite field $k$ of characteristic $\neq 2$. Let $P$ be a projective $A$-module of rank $n$ with $2n\geq d+3$. Let $I$ be an ideal of $A[T]$ of height $n$ and…

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 $R$ be an affine algebra over an algebraically closed field of characteristic $0$ with dim$(R)=n$. Let $P$ be a projective $A=R[T_1,\cdots,T_k]$-module of rank $n$ with determinant $L$. Suppose $I$ is an ideal of $A$ of height $n$ such…

We introduce a class of Stanley-Reisner ideals called generalized complete intersection, which is characterized by the property that all the residue class rings of powers of the ideal have FLC. We also give a combinatorial characterization…

Let $R$ be a polynomial ring over a field. We prove an upper bound for the multiplicity of $R/I$ when $I$ is a homogeneous ideal of the form $I=J+(F)$, where $J$ is a Cohen-Macaulay ideal and $F\notin J$. The bound is given in terms of two…

This paper is a continuation of a previous work by the author and G. Puninski where iterated intersections of powers of ideals were studied in rings of iterated differential polynomials. We present a method which can be used to show that…

This paper studies mixed multiplicities of an arbitrary standard bigraded algebra and mixed multiplicities of two ideals I, J in a local ring (A,m), where I is an m-primary ideal and J an arbitrary ideal. The main results are criteria for…

This paper deals with well-known notion of $PF$-rings, that is, rings in which principal ideals are flat. We give a new characterization of $PF$-rings. Also, we provide a necessary and sufficient condition for $R\bowtie I$ (resp., $R/I$…

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…

Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of quasi $J$-ideal which is a generalization of $J$-ideal. A proper ideal of $R$ is called a quasi $J$-ideal if its radical is a $J$-ideal. Many…