# Homework Help: Quotient Rings and Homomorphisms

1. Mar 8, 2010

### iwonde

1. The problem statement, all variables and given/known data
Let R and S be rings. Show that $$\pi$$:RxS->R given by $$\pi$$(r,s)=r is a surjective homomorphism whose kernel is isomorphic to S.

2. Relevant equations

3. The attempt at a solution
To show that $$\pi$$ is a homomorphism map, I need to show that it's closed under addition and multiplication. And to show that it's surjective, I need to show for every x in RxS, f(x)=y for y in R. What does it mean when the kernel is isomorphic to S?

2. Mar 8, 2010

### Tinyboss

You need to review your definitions of ring homomorphism and surjective function. Neither one that you gave is accurate.