Is $\mathbb Z[x^2]$ an ideal of $\mathbb Z[x]$?

Question

Here is the. Question I am wondering about:

Is $\mathbb Z[x^2]$ an ideal of $\mathbb Z[x]$?

My question is:

My professor defined $\mathbb Z[x^2]$ as the polynomials in which only even powers of $x^2$ appears, but I do not agree with him as all powers of $x^2$ I think can appear. Am I correct?

Also, I think the answer to the main question I am asking is yes, am I correct?