# Understanding elements of quotient ring.

I'm having some problems understanding what elements are in a quotient ring. I understand that they are cosets of the ideal, but when it comes to actual calculations I'm still a bit lost.

For example, consider the ring $\mathbb{Q} [x,y,z]/(x,y)$. Am I correct if I think that the quotient essentially set all terms containing $x$ or $y$ to $0$, so that we end up with simply $\mathbb{Q}[z]$ (or something isomorphic to it)?

• Yes, $\mathbb{Q}[x,y,z]/(x,y)\cong\mathbb{Q}[z]$ – Mark Oct 24 '17 at 0:41
• – lhf Oct 24 '17 at 0:44
• Yes, the quotient map sends the element $x \in \mathbb{Q}[x,y]$ to the coset $x + (x,y) = (x,y)$, which is the zero element. Same with $y$. But $z$ is sent to $z + (x,y)$. – Matthew Leingang Oct 24 '17 at 0:44

It is right, but be carefully (although not being this the case) if you have algebraic relations between elements and the ones you view as $0$ these relations involve changes on those elements. In this case the situation is simple as the elements $x,y,z$ are (algebraically) independent. Think at your ideal as a set of algebraic relations which you impose on your ring, getting the quotient with your imposed relations (and their consequences, i.e. other elements in the ideal)