Characteristic and Principal Ideal.

Question

This might be a simple question for some of you, but I am quite confused on the whole concept of principal ideals.

Question 1: What is the characteristic of $\mathbb{Z}_2[X,Y]$ where it is the ring of polynomials in $X$ and $Y$.

I know the characteristic of a ring is the least such positive integers such that $na=0$

I also know that $\mathbb{Z}_n$ has characteristic $n$, but that is about it

Question 2: Is every ideal in $\mathbb{Z}_2[X,Y]$ principal? Why?

I would really appreciate some help on this.

Answer 1

Observe that if $A$ is a subring of $B$, then $A$ and $B$ have the same characteristic, assuming the rings have unity $1$, because in this case the characteristic is the least positive integer $n$ such that $n1=0$ (if such an integer exists, otherwise it the characteristic is $0$).

Since $\mathbb{Z}_2$ is a subring of $\mathbb{Z}_2[X,Y]$, …

Consider now the ideal $I$ generated by $X$ and $Y$. If it is principal, it is $I=(f)$ for some polynomial $f(X,Y)$. This implies that $$ X=f(X,Y)g(X,Y),\quad Y=f(X,Y)h(X,Y) $$ for some polynomials $g$ and $h$. Can you get a contradiction from this? Hint: consider the degrees in $X$ and $Y$ and exploit the fact that $\mathbb{Z}_2$ is a domain, actually a field.

Answer 2

Hints:

$2\cdot1=0$, and consider $(X,Y)$.