Quotient Rings and Homomorphisms

[iwonde]

Homework Statement

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.

Homework Equations

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?

 

[Tinyboss]

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