Related papers: Is addition definable from multiplication and succ…

Let J and J' be Jordan rings. We prove under some conditions that if J contains a nontrivial idempotent, then n-multiplicative maps and n-multiplicative derivations from J to J' are additive maps.

Let $R$ be a commutative ring with identity. A unit $u$ of $R$ is called exceptional if $1-u$ is also a unit. When $R$ is a finite commutative ring, we determine the additive and multiplicative structures of its exceptional units; and then…

An associative ring $R$ with identity is left pseudo-morphic if for every $a$$\in$$R$, there exists $b$$\in$$R$ such that $Ra=l_R(b)$. If, in addition, $l_R(a)=Rb$, then $R$ is called left morphic. $R$ is morphic if it is both left and…

Let $R$ be a ring (not necessary commutative) with non-zero identity. The unit graph of $R$, denoted by $G(R)$, is a graph with elements of $R$ as its vertices and two distinct vertices $a$ and $b$ are adjacent if and only if $a+b$ is a…

Let $X$ and $Y$ be compact Hausdorff spaces, and let $C(X)$ and $C(Y)$ denote the commutative Banach algebras of all continuous complex-valued functions on $X$ and $Y$, respectively. We study bijective maps $T$ from $C(X)$ onto $C(Y)$ which…

Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices over a field $\mathbb{F}$ of characteristic not equal to $2$. If $n\ge 2$, we show that an arbitrary map $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$ is Jordan multiplicative,…

Let $R$ be a commutative ring with $\Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $\T(\Gamma(R))$. It is the (undirected) graph with all elements of $R$ as vertices, and for distinct $x, y\in…

We introduce a model-complete theory which completely axiomatizes the structure $Z_{\alpha}=(Z, +, 0, 1, f)$ where $f : x \to \lfloor{\alpha} x \rfloor $ is a unary function with $\alpha$ a fixed transcendental number. When $\alpha$ is…

A ring is *unit-additive* if a sum of units is always either a unit or nilpotent. For example, $k[X]$ and $k[X]/(X^2)$ are unit-additive, but $\mathbb Z$ is not. We prove a wide-ranging theorem about unit-additivity in semigroup rings,…

A function $f:R\to R$, where $R$ is a commutative ring with unit element, is called polyfunction if it admits a polynomial representative $p\in R[x]$. Based on this notion we introduce ring invariants which associate to $R$ the numbers…

We establish formulas for computation of the higher algebraic $K$-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let ${\mathcal C}$ be an additive category, and let $Y\ra X$ be a covariant morphism…

Let $A$ and $B$ be unital rings. An additive map $T:A\to B$ is called a weighted Jordan homomorphism if $c=T(1)$ is an invertible central element and $cT(x^2) = T(x)^2$ for all $x\in A$. We provide assumptions, which are in particular…

The purpose of this paper is to study the (different) notions of homo-derivations. These are additive mappings $f$ of a ring $R$ that also fulfill the identity \[ f(xy)=f(x)y+xf(y)+f(x)f(y) \qquad \left(x, y\in R\right), \] or (in case of…

We study whether a unital associative algebra $ A $ over a field admits a decomposition of the form $A = Z(A) + [A,A]$ where $ Z(A) $ is the center of $ A $ and $ [A,A] $ denotes the additive subgroup of $A$ generated by all additive…

By an approximate subring of a ring we mean an additively symmetric subset $X$ such that $X\cdot X \cup (X +X)$ is covered by finitely many additive translates of $X$. We prove that each approximate subring $X$ of a ring has a locally…

Let $R$ be a ring with involution containing a nontrivial symmetric idempotent element $e$. Let $\delta: R\rightarrow R$ be a mapping such that $\delta(ab)=\delta(b)a^{\ast}+b^{\ast}\delta(a)$ for all $a,b\in R$, we call $\delta$ a…

Suppose $\mathfrak{R}$ is a $2$,$3$-torsion free unital alternative ring having an idempotent element $e_1$ $\left(e_2 = 1-e_1\right)$ which satisfies $x \mathfrak{R} \cdot e_i = \{0\} \rightarrow x = 0$ $\left(i = 1,2\right)$. In this…

Let $X$ and $Y$ be compact Hausdorff spaces, $E$ and $F$ be real or complex normed spaces and $A(X,E)$ be a subspace of $C(X,E)$. For a function $f\in C(X,E)$, let $\coz(f)$ be the cozero set of $f$. A pair of additive maps $S,T: A(X,E) \lo…

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…

Let $A,B$ be C*-algebras, $B_A(0;r)$ the open ball in $A$ centered at $0$ with radius $r>0$, and $H:B_A(0;r)\to B$ an orthogonally additive holomorphic map. If $H$ is zero product preserving on positive elements in $B_A(0;r)$, we show, in…