Understanding elements of quotient ring.

Question

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, 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 00:44

score 1 · Answer 1 · answered Oct 24 '17 at 00:44

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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)

answered Oct 24 '17 at 00:44

Tancredi

1,522

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If you'd like to know more about computational aspects of the part about "and their consequences", you might find the topic of Groebner bases of ideals interesting. – Daniel Schepler Oct 24 '17 at 01:30

Understanding elements of quotient ring.

1 Answers1