Related papers: Infinite Easier Waring Constants for Commutative R…

We generalize the notion of Davenport constants to a `higher degree' and obtain various lower and upper bounds, which are sometimes exact as is the case for certain finite commutative rings of prime power cardinality. Two simple examples…

We show that a ring $\,R\,$ has two idempotents $\,e,e'\,$ with an invertible commutator $\,ee'-e'e\,$ if and only if $\,R \cong {\mathbb M}_2(S)\,$ for a ring $\,S\,$ in which $\,1\,$ is a sum of two units. In this case, the…

Let $A_1,\ldots,A_s$ be unitary commutative rings which do not have non-trivial idempotents and let $A=A_1\oplus\cdots\oplus A_s$ be their direct sum. We describe all idempotents in the $2\times 2$ matrix ring $M_2(A[[X]])$ over the ring…

Let R be an associative ring.In the paper we study n-generalized commutators of rings and prove that if R is a noncommutative prime ring and n > 2, then every nonzero n-generalized Lie ideal of R contains a nonzero ideal. Therefore, if R is…

We prove that for all integers $k \geq 1$, there exists a constant $C_k$ depending only on $k$ such that for all $q > C_k$ and for all $n \geq 1$ every matrix in $M_n(\mathbb F_q)$ is a sum of two $k$th powers.

In this paper, we introduce a new class of rings calling them {\it 2-UNJ rings}, which generalize the well-known 2-UJ, 2-UU and UNJ rings. Specifically, a ring $R$ is called 2-UNJ if, for every unit $u$ of $R$, the inclusion $u^2 \in 1 +…

Let $R$ be a commutative ring with identity. An element $r \in R$ is said to be absolutely irreducible in $R$ if for all natural numbers $n>1$, $r^n$ has essentially only one factorization namely $r^n = r \cdots r$. If $r \in R$ is…

Through this paper, we study the rings in which every unit's square is an element of the set $1+\sqrt{J(R)}$, and call them $2-\sqrt{J}U$ rings. Here, $\sqrt{J(R)}=\{x \in R: x^m \in J(R)$ for some $m \geq 1 \}$. We show that every…

We characterize extensions of commutative rings $R\subset S$ such that $R\subset T$ is minimal for each $R$-subalgebra $T$ of $S$ with $T\neq R,S$. This property is equivalent to $R\subset S$ has length 2. Such extensions are either…

Let $R$ be a commutative unitary ring. An idempotent in $R$ is an element $e\in R$ with $e^2=e$. The Erd\H{o}s-Burgess constant associated with the ring $R$ is the smallest positive integer $\ell$ (if exists) such that for any given $\ell$…

The aim of this paper is to study the probability that the commutator of an arbitrarily chosen pair of elements, each from two different subrings of a finite non-commutative ring equals a given element of that ring. We obtain several…

A cover of a unital, associative (not necessarily commutative) ring $R$ is a collection of proper subrings of $R$ whose set-theoretic union equals $R$. If such a cover exists, then the covering number $\sigma(R)$ of $R$ is the cardinality…

A ring is called a commutator ring if every element is a sum of additive commutators. In this paper we give examples of such rings. In particular, we show that given any ring R, a right R-module N, and a set X, End_R(\bigoplus_X N) and…

Jacobson's commutativity theorem says that a ring is commutative if, for each $x$, $x^n = x$ for some $n > 1$. Herstein's generalization says that the condition can be weakened to $x^n-x$ being central. In both theorems, $n$ may depend on…

Let $R$ be any associative ring with $1$, $n\ge 3$, and let $A,B$ be two-sided ideals of $R$. In the present paper we show that the mixed commutator subgroup $[E(n,R,A),E(n,R,B)]$ is generated as a group by the elements of the two following…

Zeckendorf's theorem states that every positive integer can be uniquely decomposed into nonadjacent Fibonacci numbers. On the other hand, Chung and Graham proved that every positive integer can be uniquely written as a sum of even-indexed…

Let $R$ be a commutative ring with $1\neq 0$ and $n$ be a fixed positive integer. A proper ideal $I$ of $R$ is said to be an \textit{$n$-OA ideal} if whenever $a_1a_2\cdots a_{n+1}\in I$ for some nonunits $a_1,a_2,\ldots,a_{n+1}\in R$, then…

It is known that for an arbitrary positive integer \(n\) the sequence \(S(x^n)=(1^n, 2^n, \ldots)\) is complete, meaning that every sufficiently large integer is a sum of distinct \(n\)th powers of positive integers. We prove that every…

Let $R$ be an associative ring with unity $1$ and consider that $2,k$ and $2k\in \mathbb{N}$ are invertible in $R$. For $m\geq 1$ denote by $UT_n(m,R)$ and $UT_{\infty}(m,R)$, the subgroups of $UT_n(R)$ and $UT_{\infty}(R)$ respectively,…

We know that for a finite field $F$, every function on $F$ can be given by a polynomial with coefficients in $F$. What about the converse? i.e. if $R$ is a ring (not necessarily commutative or with unity) such that every function on $R$ can…