Related papers: Powers Vs. Powers

This paper explores the duality between ideals of the ring $B_1(X)$ of all real valued Baire one functions on a topological space $X$ and typical families of zero sets, called $Z_B$-filters, on $X$. As a natural outcome of this study, it is…

Let J \subseteq I be ideals in a commutative Noetherian ring R, and r,s \geq 0. We say that J is a demotion of I if I^r J^s = I^{r+s} \cap J^s for all r,s \geq 0. In this paper, we mainly aim to explore this notion in polynomial rings. In…

We study the "q-commutative" power series ring R:=k_q[[x_1,...,x_n]], defined by the relations x_ix_j = q_{ij}x_j x_i, for multiplicatively antisymmetric scalars q_{ij} in a field k. Our results provide a detailed account of prime ideal…

Let R be a commutative ring and I an ideal of R. A sub-ideal J of I is a reduction of I if JI^n = I^n+1 for some positive integer n. The ring R has the (finite) basic ideal property if (finitely generated) ideals of R do not have proper…

Let $(R,\mathfrak m,\mathsf k)$ be either a fiber product or an artinian stretched Gorenstein ring, with $\operatorname{ch}(\mathsf k)\neq 2$ in the latter case. We prove that the ideals of minors of the minimal free resolution of any…

Let $A$ and $B$ be standard graded polynomial rings over a field $k$ and $I$ and $J$ be non-zero, proper homogeneous ideals contained in $A$ and $B$, respectively. Denote by $P$ the sum of $I$ and $J$ in $R=A\otimes_k B$. Under reasonable…

Let $k$ be a field, let $A$ and $B$ be polynomial rings over $k$, and let $S= A \otimes_k B$. Let $I \subseteq A$ and $J \subseteq B$ be monomial ideals. We establish a binomial expansion for rational powers of $I+J \subseteq S$ in terms of…

In this paper, we introduce the concept of S-J-ideals in both commutative and noncommutative rings. For a commutative ring R and a multiplicatively closed subset S, we show that many properties of J-ideals apply to S-J-ideals and examine…

If $R$ is a valuation domain of maximal ideal $P$ with a maximal immediate extension of finite rank it is proven that there exists a finite sequence of prime ideals $P=L_0\supset L_1\supset...\supset L_m\supseteq 0$ such that…

Let I be a complete m-primary ideal of a regular local ring (R,m). In the case where R has dimension two, the beautiful theory developed by Zariski implies that I factors uniquely as a product of powers of simple complete ideals and each of…

Let $R$ be a Noetherian local ring and $m$ a positive integer. Let $I$ be the ideal of $R$ generated by the maximal minors of an $m \times (m + 1)$ matrix $M$ with entries in $R$. Assuming that the grade of the ideal generated by the…

We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then…

A C*-algebra $A$ is said to be stable if it is isomorphic to $A \otimes K(\ell_2)$. Hjelmborg and R\o rdam have shown that countable inductive limits of separable stable C*-algebras are stable. We show that this is no longer true in the…

We prove the following;Theorem:Let R be a prime noetherian ring with k.dimR = n, n a finite non-negative integer. We refer the reader to the definitions (1.1) of this paper.For a fixed non-negative integer m, m<n let Xm be the full set of…

Given an ideal $I$ in a Noetherian ring, one can ask the containment question: for which $m$ and $r$ is the symbolic power $I^{(m)}$ contained in the ordinary power $I^r$? C. Bocci and B. Harbourne study the containment question in a…

A pair of elements $a,b$ in an integral domain $R$ is an idempotent pair if either $a(1-a) \in bR$, or $b(1-b) \in aR$. $R$ is said to be a PRINC domain if all the ideals generated by an idempotent pair are principal. We show that in an…

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…

Let R be an integral domain and I a nonzero ideal of R. A sub-ideal J of I is a t-reduction of I if (JI^{n})_{t}=(I^{n+1})_{t} for some positive integer n. An element x in R is t-integral over I if there is an equation x^{n} + a_{1}x^{n-1}…

Let $H$ be a connected Hopf algebra acting on an algebra $A$. Working over a base field having characteristic $0$, we show that for a given prime (semi-prime, completely prime) ideal $I$ of $A$, the largest $H$-stable ideal of A contained…

This paper is concerned with tight closure in a commutative Noetherian ring $R$ of prime characteristic $p$, and is motivated by an argument of K. E. Smith and I. Swanson that shows that, if the sequence of Frobenius powers of a proper…