Ring homomorphism
In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function f : R → S that preserves addition, multiplication and multiplicative identity; that is, f ( a + b ) = f ( a ) + f ( b ) , f ( a b ) = f ( a ) f ( b ) , f ( 1 ) = 1 , {\displaystyle {\begin{aligned}f(a+b)&=f(a)+f(b),\\f(ab)&=f(a)f(b),\\f(1)&=1,\end{aligned}}} for all a, b in R. These conditions imply that additive inverses and the additive identity are also preserved (see Group homomorphism).