martindale proved that under some conditions every multiplicative isomorphism between two rings is additive. in this paper, we extend this theorem to a larger class of mappings and conclude that every multiplicative $(alpha, beta)-$derivation is additive.

Martindale proved that under some conditions every multiplicative isomorphism between two rings is additive. In this paper, we extend this theorem to a larger class of mappings and conclude that every multiplicative $(alpha, beta)-$derivation is additive.

hence it yields an isomorphism between the quotient additive group C/2πiZ and the multiplicative group C×. The group C× is the group of complex points of the multiplicative group Gm; z 7→ e is the exponential function of the multiplicative group Gm. We shall replace this algebraic group by an elliptic curve. We could replace it also by other commutative algebraic groups. As a first example, the...

Let $\Re$ and $\Re'$ unital $2$,$3$-torsion free alternative rings $\varphi: \Re \rightarrow \Re'$ be a surjective Lie multiplicative map that preserves idempotents. Assume has nontrivial Under certain assumptions on $\Re$, we prove $\varphi$ is of the form $\psi + \tau$, where $\psi$ either an isomorphism or negative anti-isomorphism onto $\tau$ additive mapping into centre which maps commutat...

The aim of this paper is to show that under some mild conditions a functional equation of multiplicative $(alpha,beta)$-derivation is superstable on standard operator algebras. Furthermore, we prove that this generalized derivation can be a continuous and an inner $(alpha,beta)$-derivation.

A large class of multiplicative submonoids of the natural numbers is presented, which includes congruence monoids as well as numerical monoids (by isomorphism). For monoids in this class, the important factorization property of finite elasticity is characterized.

let $l$ be a completely regular frame and $mathcal{r}l$ be the ringof continuous real-valued functions on $l$. we study the frame$mathfrak{o}(min(mathcal{r}l))$ of minimal prime ideals of$mathcal{r}l$ in relation to $beta l$. for $iinbeta l$, denoteby $textit{textbf{o}}^i$ the ideal${alphainmathcal{r}lmidcozalphain i}$ of $mathcal{r}l$. weshow that sending $i$ to the set of minimal prime ideals...

A large class of multiplicative submonoids of the natural numbers is presented, which includes congruence monoids as well as numerical monoids (by isomorphism). For monoids in this class, the important factorization property of finite elasticity is characterized.

In this paper, we address the additivity of n-multiplicative isomorphisms and derivations on gamma rings. We proved that, if $${{\mathfrak {M}}}\, $$ is a $$\Gamma -ring satisfying some conditions, then any isomorphism $$\left( \varphi , \phi \right) onto an arbitrary ring additive.