Is addition definable from multiplication and successor?

Abstract

A map f : R → S {f\colon R\to S} between (associative, unital, but not necessarily commutative) rings is a brachymorphism if f ⁢ ( 1 + x ) = 1 + f ⁢ ( x ) {f(1+x)=1+f(x)} and f ⁢ ( x ⁢ y ) = f ⁢ ( x ) ⁢ f ⁢ ( y ) {f(xy)=f(x)f(y)} whenever x , y ∈ R {x,y\in R} . We tackle the problem whether every brachymorphism is additive (i.e., f ⁢ ( x + y ) = f ⁢ ( x ) + f ⁢ ( y ) {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 × 2 {2\times 2} matrices over a commutative ring; • R is Engelian; • every element of R is a sum of π-regular and central elements (this applies to π-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 ] {K[M]} for a commutative ring K and a π-regular monoid M; • R is the Weyl algebra 𝖠 1 ⁢ ( K ) {\mathsf{A}_{1}(K)} over a commutative ring K with positive characteristic; • f is the power function x ↦ x n {x\mapsto x^{n}} over any ring; • f is the determinant function over any ring R of n × n {n\times n} matrices, with n ≥ 3 {n\geq 3} , over a commutative ring, such that if n > 3 {n>3} , then R contains n scalar matrices with non zero divisor differences.