Ring theory questions I was doing exercises of algebra and I’m struggling with this exercise. I have to say if the propositions are true or false, demonstrate it if it is true and give a counterexample if it is false. Here are the questions.

Let $P \in \mathbb{C}[X,Y]$ be an irreducible polynomial. The quotient ring $\mathbb{C}[X,Y]/\langle P\rangle$ is a field.

I think I can consider $\mathbb{C}[X,Y]$ as $\mathbb{C}[X][Y]$ and then do the usual proof. So I would say this is true.

Let $I$ be a proper ideal of a commutative ring $A$. If all the elements of $A / I $ are invertible then $I$ is a maximal ideal of $A$.

I know that $I$ is maximal iff $A/ I$ is a field.  So I would say this is false since $\overline0_A \in A / I $ is invertible thus $A / I $ cannot be a field.

In a commutative ring $A$, the product of two divisors of zero is a divisor of zero.

The definition we have if this: $a\in A \setminus \{0_A\}$ is a divisor of zero if $\exists b\in A \setminus\{0_A\}$ such that $ab=0_A$. I would say that this is true
Correct me if I did mistakes thank you!
 A: *

*The polynomial $X\in\Bbb{C}[X,Y]$ is irreducible, but $\Bbb{C}[X,Y]/(X)$ is not a field.

*The quotient $A/I$ is not the same as the complement $A\backslash I$.

*Do you think that it is false or true? You write both. Anyway, note that if $a$ is a zero divisor then there exists some nonzero $b$ such that $ab=0$. It is not true that $ab=0$ for all $b\in A$. So your argument does not hold.

A: Question 1. If a polynomial $P$ in the ring $\mathbb{C}[X]$ is irreducible, then $\langle P\rangle$ is a maximal ideal. This follows from the fact that every ideal in $\mathbb{C}[X]$ is principal (instead of $\mathbb{C}$ you can have any field). The statement is no longer valid in $\mathbb{C}[X,Y]$, which is not a principal ideal domain. Consider $\langle X\rangle\subsetneq\langle X,Y\rangle\subsetneq\mathbb{C}[X,Y]$.
The ideal generated by an irreducible polynomial is maximal in the set of proper principal ideals (the proof of the single variable case carries over), but not in the set of all proper ideals. Actually, no irreducible polynomial in $\mathbb{C}[X,Y]$ generates a maximal ideal.
Question 2. Is a ring $R$ in which every element is invertible a field? Also $0$ should be invertible, so…
Question 3. Let $a$ be a zero-divisor, so $a\ne0$ and $ab=0$ for some $b\ne0$. Then also $b$ is a zero-divisor, isn't it? But is the product $ab$ a zero-divisor?
