Abstract Algebra/Ring Homomorphisms
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[edit] Definition
Let R and S be two rings. Then a function 
is called a ring homomorphism if for every 
, the following properties hold:
- f(r1r2) = f(r1)f(r2),
- f(r1 + r2) = f(r1) + f(r2).
In other words, f is a ring homomorphism if it preserves additive and multiplicative structure.
Furthermore, if R and S are rings with unity and f(1R) = 1S, then f is called a unital ring homomorphism.
Example: Let f be a function from the ring Z to the ring of 2-by-2 matrices with its first element being an element in Z, and 0 elsewhere. Define f(z) simply to be the matrix with z as its first element, and 0 elsewhere. Then one can easily see that it is a homomorphism, but not a unital ring homomorphism.
