Some possible typos in solution of 9.1.12 Dummit & Foote.

Question

Here is the question I was trying to solve:

Let $R = \mathbb Q[x,y,z]$ and let bars denote passage to $ \mathbb Q[x,y,z] / (xy - z^2).$ Prove that $\overline{P} = (\overline{x}, \overline{z})$ is a prime ideal. Show that $\overline{xy} \in \overline{P}^2$ but that no power of $\bar{y}$ lies in $\overline{P}^2.$

I was reading the following question solution here:

$R = \mathbb{Q}[x,y,z]$ and let bars denote passage to $\mathbb{Q}[x,y,z]/(xy-z^2)$ Prove $\bar{P} = (\bar{x},\bar{z} )$

The solution is mixing the $x$ and $y$ in some places. Like, in the following line:

"Let $P = (x,z)$ it's clear that $xy \in (x,z)$ and $z^2 \in (x,z)$ thus $ xy-z^2 \in (x,z)$ further $\overline{(xy-z^2)} \subseteq (\bar{x},\bar{y})$"

1- should not the last ideal be $(\bar{x},\bar{z})$ instead?

2- Also, did the authur intentially meant to say $xy \in (x,z)$? Is $xy \in (x,z)$ because the ideal is closed under multiplication by the elements of the ring?

Also, in the following line:

"$(\bar{x},\bar{z})/\overline{(xy-z^2)}$ is an ideal of $\mathbb{Q}[x,y,z]/(xy-z^2)$."

Why the ideal $(\bar{x},\bar{z})/\overline{(xy-z^2)}$ looks like this? should not it be called $(\bar{x},\bar{z})$ only? what is the differnce between the bar over x and z and that over $(xy-z^2)$?

can someone explain these points for me please?

Answer 1

⦁ Answering your 2nd question, $xy∈(x,z)$ because $xy$ can be written as $xy=x\cdot{y}+z\cdot0$ and $x\cdot{y}+z\cdot0∈(x,z)$ since $y,0∈\Bbb{Q}[x,y,z]$. Thus, indeed $xy∈(x,z)$.

⦁ Answering your 1st question, the bar notation here denotes the passage defined by the canonical ring homomorphism $f∶\Bbb{Q}[x,y,z]→\Bbb{Q}[x,y,z]/(xy-z^2)$ with $ker⁡f=(xy-z^2)$, so that $\overline{(xy-z^2)}=\overline{ker⁡f}=\overline{0}$, the zero ideal in the quotient ring $\Bbb{Q}[x,y,z]/(xy-z^2)=\overline{\Bbb{Q}[x,y,z]}.$ And the zero ideal $\overline{0}=\overline{(xy-z^2)}$ is contained in every ideal of $\overline{\Bbb{Q}[x,y,z]}$, namely, in $(\overline{x},\overline{y})$, $(\overline{x},\overline{z})$, ... . So, the initial author's statement "$\overline{(xy-z^2)}⊆(\overline{x},\overline{y})$" is obvious without almost any preceding argument and doesn't make much sense here.

⦁ Answering your last question, $(\overline{x},\overline{z})$ is already an ideal of $\Bbb{Q}[x,y,z]/(xy-z^2)=\overline{\Bbb{Q}[x,y,z]}$, because $\overline{x},\overline{z}∈\overline{\Bbb{Q}[x,y,z]}$. For the same reason as above, considering its quotient "$(\overline{x},\overline{z})/\overline{(xy-z^2)}$" by the zero ideal $\overline{0}=\overline{(xy-z^2)}$ doesn't make much sense, because this would mean the passage from our quotient ring $\Bbb{Q}[x,y,z]/(xy-z^2)=\overline{\Bbb{Q}[x,y,z]}$ to the quotient ring by $\overline{0}=\overline{(xy-z^2)}$ of this quotient ring: $\overline{\Bbb{Q}[x,y,z]}/\overline{(xy-z^2)}=\overline{\Bbb{Q}[x,y,z]}/\overline{0}≅\overline{\Bbb{Q}[x,y,z]}$, so that we get the isomorphism of our quotient ring $\Bbb{Q}[x,y,z]/(xy-z^2)=\overline{\Bbb{Q}[x,y,z]}$ to itself; doesn't make much sense.
So, yes, it should be "$(\overline{x},\overline{z})$ is an ideal of $\Bbb{Q}[x,y,z]/(xy-z^2)$".

⦁ The difference between the notations "$(\overline{x},\overline{z})$" and "$\overline{(xy-z^2)}$" is as follows: $(\overline{x},\overline{z})=(x+kerf,z+kerf)=(x+(xy-z^2),z+(xy-z^2))$
$\overline{(xy-z^2)}=(xy-z^2)/kerf=(xy-z^2)/(xy-z^2)=\overline{0}$,
if this is what you meant in your last question.

⦁⦁⦁ I think all of these possible "typos" arose, because the Third Isomorphism Theorem was applied not entirely correct in the original post. There should have been something like the following:
Since $xy-z^2=xy+z(-z)$, where $y,-z∈\Bbb{Q}[x,y,z]$, so that $xy+z(-z)∈(x,z)$, it follows that $xy-z^2∈(x,z)$. Hence $(xy-z^2)⊂(x,z)$ (the containment is proper, since, for example, $z∉(xy-z^2)$). Now the Third Isomorphism Theorem for Rings can be applied: $$\frac{\Bbb{Q}[x,y,z]/(xy-z^2)}{(x,z)/(xy-z^2)} ≅ \frac{\Bbb{Q}[x,y,z]}{(x,z)} ≅\Bbb{Q}[y]$$ Since $\Bbb{Q}$ is an integral domain, $\Bbb{Q}[y]$ is an integral domain, therefore, $(x,z)/(xy-z^2)$ is a prime ideal of $\Bbb{Q}[x,y,z]/(xy-z^2)$. Denoting $P=(x,z)$ and using the bar notation as before, we get that $P/(xy-z^2)=(x,z)/(xy-z^2)$ ⇔ $\overline{P}=\overline{(x,z)}$ is a prime ideal of $\Bbb{Q}[x,y,z]/(xy-z^2)=\overline{\Bbb{Q}[x,y,z]}$. Since the ring homomorphism $f$ (given above) is surjective, and $ker⁡f=(xy-z^2)$ is contained in $(x,z)$, it follows that $\overline{(x,z)}=(\overline{x},\overline{z})$ [it is not hard to prove using the mutual inclusion pattern]. Thus, $\overline{P}= (\overline{x},\overline{z})$ is a prime ideal of $\Bbb{Q}[x,y,z]/(xy-z^2)=\overline{\Bbb{Q}[x,y,z]}$.

P.S. Hope this helps.