Lecture 7: Homomorphism

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.

A homomorphism between two rings (R,+, •) and (R ,+ , • ) is a function f : R ? R which preserves both ring operations. The definition can be given as follows:
Definition 1. Let (R,+, •) and (R ,+ , • ) be two rings and f is a function from R into R (i.e function f : R ? R ). Then f is said to be a (ring) homomorphism from R,+, •) into (R ,+ , • ) iff
1. f(a+b)=f(a)+ f(b);
2. f(a•b)=f(a) • f(b), for all a,b in R.

Example 1. Let (R,+, •) and (R ,+ , • ) be arbitrary rings and f : R ? R be the function that maps each element of R onto the zero element 0 of (R ,+ , • ). Show that f is a ring homomorphism?

Solution: Based on Definition of homomorphism , f is a ring homomorphism, since
1. f(a+b)=0 =0 +0 = f(a)+ f(b); and
2. f(a•b)= 0 =0 • 0 = f(a) • f(b), for all a,b in R.

The function f in this case is called trivial homomorphism.