Is addition definable from multiplication and successor?
Is addition definable from multiplication and successor?
A map $f\colon R\to S$ between (associative, unital, but not necessarily commutative) rings is a \emph{brachymorphism} if $f(x+1)=f(x)+1$ 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: $\bullet$ $R$ is finite (or, …