Hey guys.

I have a commutative ring R, with two ideals, I and J, in R, where I $\displaystyle \subseteq$ J.

Now I have to show that $\displaystyle \varphi : R / I \rightarrow R / J $ given by $\displaystyle \varphi ( x + I ) = ( x + J ) $ is a well defined, surjective ring homomorphism.

Right now I cant even figure out what i need to do, to prove that it is well-defined. I'm missing parts of what I should have learned, but as far as I've been able to figure out, a homomorphism is well-defined when for any $\displaystyle a,b \in R/I$, $\displaystyle a = b \Leftrightarrow \varphi (a) = \varphi (b)$.

But I can't figure out how to prove that. If someone could give me a hint, I would really appreciate it. Perhaps I need a more stringent definition of "well-defined" or if you could give me an example of a proof kind of like mine?

All help is appreciated.

Morten