Is addition definable from multiplication and successor?

Abstract

A map $f:R\to S$ between (associative, unital, but not necessarily commutative) rings is a\emph{brachymorphism} if $f(1+x)=1+f(x)$ and $f(xy)=f(x)f(y)$ whenever $x,y\in R$. We tackle the problem whether every brachymorphism is additive (i.e., $f(x+y)=f(x)+f(y)$), showing that in many contexts, including the following, the answer is positive: $R$ is finite (or, more generally, $R$ is left or right Artinian); $R$ is any ring of $2\times 2$ matrices over a commutative ring; $R$ is Engelian; every element of $R$ is a sum of $\pi$-regular and central elements (this applies to $\pi$-regular rings, Banach algebras, and power series rings); $R$ is the full matrix ring of order greater than $1$ over any ring; $R$ is the monoid ring $K[M]$ for a commutative ring $K$ and a $\pi$-regular monoid $M$; $R$ is the Weyl algebra $A_1(K)$ over a commutative ring $K$ with positive characteristic; $f$ is the power function $x\mapsto x^n$ over any ring; $f$ is the determinant function over any ring $R$ of $n\times n$ matrices, with $n\ge 3$, over a commutative ring, such that if $n>3$ then $R$ contains $n$ scalar matrices with non zero divisor differences.