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)?

  • $\begingroup$ Yes, $\mathbb{Q}[x,y,z]/(x,y)\cong\mathbb{Q}[z]$ $\endgroup$ – Mark Oct 24 '17 at 0:41
  • $\begingroup$ See also math.stackexchange.com/a/1817020/589 $\endgroup$ – lhf Oct 24 '17 at 0:44
  • $\begingroup$ 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 $z + (x,y)$. $\endgroup$ – 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)

  • 1
    $\begingroup$ 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. $\endgroup$ – Daniel Schepler Oct 24 '17 at 1:30

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