Related papers: On a theorem of Stafford

Two classical results of Stafford say that every (left) ideal of the $n$-th Weyl algebra $A_n$ can be generated by two elements, and every holonomic $A_n$-module is cyclic, i.e. generated by one element. We modify Stafford's original proofs…

For the algebra $\mI_1= K<x, \frac{d}{dx}, \int>$ of polynomial integro-differential operators over a field $K$ of characteristic zero, a classification of simple modules is given. It is proved that $\mI_1$ is a left and right coherent…

Let $K$ be a unital associative and commutative ring and let $K \langle X \rangle$ be the free unital associative $K$-algebra on a non-empty set $X$ of free generators. Define a left-normed commutator $[a_1, a_2, \dots , a_n]$ inductively…

Let R be a Stanley-Reisner ring (that is, a reduced monomial ring) with coefficients in a domain k, and K its associated simplicial complex. Also let D_k(R) be the ring of k-linear differential operators on R. We give two different…

It is well-known that each left ideals in a matrix rings over a finite field is generated by an idempotent matrix. In this work we compute the number of left ideals in these rings, the number of different idempotents generating each left…

We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number $N$ such that every element is a sum of $N$ products of pairs of commutators. We show that one can take $N \leq 2$ for…

Let $A_{1} := k [t, \partial ]$ be the first algebra over a field $k$ of characteristic zero. One can associate to each right ideal $I$ of $A_1$ its Stafford subgroup, which is a subgroup of $\Aut_k(A_1)$, the automorphism group of the ring…

A theorem of Cohen from 1950 states that a commutative ring is Noetherian if and only if every prime ideal is finitely generated. In this note, we establish analogues of this result in tensor triangular geometry. In particular, for an…

In 1965, Baxter established that a simple ring is either a field or that every one of its elements can be expressed as a sum of products of commutator pairs. In a recent paper, Gardella and Thiel demonstrated that every element in a…

Let $R$ be a commutative ring with identity and let $I$ be a two-generated ideal of $R$. We denote by $\operatorname{SL}_2(R)$ the group of $2 \times 2$ matrices over $R$ with determinant $1$. We study the action of $\operatorname{SL}_2(R)$…

Based on the structure theory of pairs of skew-symmetric matrices, we give a conjecture for the Hilbert series of the exterior algebra modulo the ideal generated by two generic quadratic forms. We show that the conjectured series is an…

Given a right ideal $I$ in a ring $R$, the idealizer of $I$ in $R$ is the largest subring of $R$ in which $I$ becomes a two-sided ideal. In this paper we consider idealizers in the second Weyl algebra $A_2$, which is the ring of…

We answer a question raised by Lanier about the possibility of generating $A_n$ and $S_n$ with two elements of order $k$, where $n \geqslant k \geqslant 3$. We show that this can always be done apart from some clear exceptions.

We prove a one-to-one correspondence between the operadic ideals of the operad $\As$ and $T$-ideals. As a consequence, we show that $\As$ is noetherian and that every proper operadic ideal of $\ias$ is generated by a single element.

It was shown by Cellini and Papi that an ad-nilpotent ideal determines certain element of the affine Weyl group, and that there is a bijection between the ad-nilpotent ideals and the integral points of a simplex with rational vertices. We…

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…

Let $R$ be a polynomial ring in $N$ variables over an arbitrary field $K$ and let $I$ be an ideal of $R$ generated by $n$ polynomials of degree at most 2. We show that there is a bound on the projective dimension of $R/I$ that depends only…

The aim of the papers is to describe the left regular left quotient ring ${}'Q(R)$ and the right regular right quotient ring $Q'(R)$ for the following algebras $R$: $\mS_n=\mS_1^{\t n}$ is the algebra of one-sided inverses, where…

This paper investigates situations where a property of a ring can be tested on a set of "prime right ideals." Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff…

A classical theorem of Forster asserts that a finite module $M$ of rank $\leq n$ over a Noetherian ring of Krull dimension $d$ can be generated by $n + d$ elements. We prove a generalization of this result, with "module" replaced by…