Is addition definable from multiplication and successor?

A map <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>R</m:mi> <m:mo>→</m:mo> <m:mi>S</m:mi> </m:mrow> </m:mrow> </m:math> {f R→ S} between (associative, unital, but not necessarily commutative) rings is a brachymorphism if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> {f(1+x)=1+f(x)} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>⁢</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {f(xy)=f(x)f(y)} whenever <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mi>R</m:mi> </m:mrow> </m:math> {x,y∈ R} . We tackle the problem whether every brachymorphism is additive (i.e., <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>+</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>×</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {2× 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>K</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> {K[M]} for a commutative ring K and a π-regular monoid M ; • R is the Weyl algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>𝖠</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {A₁(K)} over a commutative ring K with positive characteristic; • f is the power function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>x</m:mi> <m:mo>↦</m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:math> {x↦ xⁿ} over any ring; • f is the determinant function over any ring R of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>×</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> {n× n} matrices, with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> {n≥ 3} , over a commutative ring, such that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>></m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> {n>3} , then R contains n scalar matrices with non zero divisor differences.