Fix a field $\displaystyle k$ and consider the subring $\displaystyle k[x^2,x^3] \subseteq k[x]$ (the former ring is all polynomials with zero coefficient in front of x). Prove that $\displaystyle k[x^2,x^3]$ is not a principal ideal domain.

I just need to find an ideal $\displaystyle I$ that is not principal, right?

$\displaystyle I=(x^2,x^3)= \{ $polynomials in $\displaystyle x$ s.t. $\displaystyle a_0=a_1=0 \}$

Is this actually true? $\displaystyle (x^2,x^3)= \{ $polynomials in $\displaystyle x$ s.t. $\displaystyle a_0=a_1=0 \}$

Is $\displaystyle I$ principal?