Related papers: Commuting and product-zero probability in finite r…

A Lie algebra $L$ over a field $\mathbb{F}$ is said to be zero product determined (zpd) if every bilinear map $f:L\times L\to \mathbb{F}$ with the property that $f(x,y)=0$ whenever $x$ and $y$ commute is a coboundary. The main goal of the…

We study a commutation pattern in which two affine families commute completely across the two families while each family retains internal noncommutativity. For one-dimensional affine groups over finite commutative rings, we prove a…

We define a random commuting $d$-tuple of $n$-by-$n$ matrices to be a random variable that takes values in the set of commuting $d$-tuples and has a distribution that is a rapidly decaying continuous weight on this algebraic set. In the…

We give an elementary proof of a result which is not as well known as it should be: a ring with a specified finite number of zero divisors is finite, with a precise bound on its order.

In [6], a constraint on invariant measures of bi-permutative cellular automata has been observed: fixed values at the positive indices determine almost-surely a uniform conditional probability on the subset of values of positive conditional…

Let $G$ be a finite simple group. In this paper we consider the existence of small subsets $A$ of $G$ with the property that, if $y \in G$ is chosen uniformly at random, then with high probability $y$ invariably generates $G$ together with…

Let R0 be a commutative associative ring (not necessarily unital), G a group and alpha a partial action by ideals that contain local units. We show that R0 is maximal commutative in the partial skew group ring R0*G if and only if R0 has the…

Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$, and assume that the characteristic of $k$ is zero or a pretty good prime for $G$. Let $P$ be a parabolic subgroup of $G$ and let $\mathfrak p$ be the…

Let $G$ be a compact connected Lie group and $n\geqslant 1$ an integer. Consider the space of ordered commuting $n$-tuples in $G$, $Hom(\mathbb{Z}^n,G)$, and its quotient under the adjoint action,…

We describe the prime ideals and, in particular, the maximal ideals in products $R = \prod D_\lambda$ of families $(D_\lambda)_{\lambda \in \Lambda}$ of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the…

One of the many equivalent formulation of the K\"othe's conjecture is the assertion that there exists no ring which contains two nil right ideals whose sum is not nil. We discuss several consequences of an observation that if the Koethe…

For a finite loop $Q$, let $P (Q)$ be the set of elements that can be represented as a product containing each element of $Q$ precisely once. Motivated by the recent proof of the Hall-Paige conjecture, we prove several universal…

Let $A$ be any unital associative, possibly non-commutative ring and let $p$ be a prime number. Let $E(A)$ be the ring of $p$-typical Witt vectors as constructed by Cuntz and Deninger and $W(A)$ be the abelian group constructed by…

Adapting Reiher's proof of Kemnitz's conjecture, we obtain two refinements of a theorem of Schmid and Zhuang. Our main results provide improved upper bounds for the Erd\H{o}s-Ginzburg-Ziv constant of rank-two-like $p$-groups, and their…

We prove several results about p-divisible groups and Rapoport-Zink spaces. Our main goal is to prove that Rapoport-Zink spaces at infinite level are naturally perfectoid spaces, and to give a description of these spaces purely in terms of…

A well known conjecture in the theory of transformation groups states that if p is a prime and (Z/p)^r acts freely on a product of k spheres, then r is less than or equal to k. We prove this assertion if p is large compared to the dimension…

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 $RG$ denote the group ring of the torsion group $G$ over a commutative ring $R$ with identity. In this paper we present proofs of some statements that appear without to be proved in the literature. We establish the valid implications…

We construct a commutative version of the group ring and show that it allows one to translate questions about the normal generation of groups into questions about the generation of ideals in commutative rings. We demonstrate this with an…

We consider random walk on a finite group $G$ as follows. We can consider $G$ as a group of substitutions. Randomly (i.e. with probability $U(g)=|G|^{-1}$ ) we choose a substitution $g \in G$ and execute it twice in a row, i.e. execute a…