Is addition definable from multiplication and successor?

TL;DR

This paper investigates whether certain ring homomorphisms that preserve addition and multiplication are necessarily additive, establishing conditions under which additivity follows in various algebraic structures.

Contribution

The paper proves that brachymorphisms are additive in many classes of rings, including matrix rings, Artinian rings, and rings with specific element decompositions, extending understanding of ring homomorphisms.

Findings

Brachymorphisms are additive in finite and Artinian rings.

Additivity holds for matrix rings over commutative rings.

Power functions and determinants are additive under specified conditions.

Abstract

A map $f\colon R\to S$ 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 $2\times2$ 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 $n\geq3$, over a commutative ring, such that if $n>3$ then $R$ contains $n$ scalar matrices with non zero divisor differences.