TITLE: Is there a typo in this question?
QUESTION [0 upvotes]: This question is from a commutative algebra assignment that I am trying to solve.

Let $k$ be a field and $X,Y,Z$ be variables. Set $R=k[X,Y,Z]/(X^2 -Y^3-1, XZ-1)$ and let $x,y,z \in R$ be the images of $X,Y,Z$ respectively. Set $t=x+z$. Let $S= k[t]$. Prove that $x,y$ are integral over $S$.

The question is in line "let $x,y,z \in R$ be the images of X,Y,Z respectively". Under what function is the image of X,Y,Z? I don't think that the function is mentioned and  I think that is a typo.

Can you please confirm?

REPLY [3 votes]: Presumably $x,y,z$ are the images of $X,Y,Z$ under the canonical quotient map $k[X,Y,Z] \to k[X,Y,Z]/(X^2 - Y^3 - 1,XZ - 1)$. Notice that it says $x,y,z \in R$ rather than $x,y,z \in k[X,Y,Z]$.
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