I was just wondering whether there would exist an element $x^{2}$ and how would the elements in general look like?
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"How does an ideal generated by $(x^2)$"... *do what*? – Mar 15 '20 at 02:32

3Also: in what ring are we? – Mar 15 '20 at 02:32
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The ideal generated by $x^2$ in $\Bbb Z[x]$ is the set of all polynomials which may be written as $x^2f(x)$ for some polynomial $f\in\Bbb Z[x]$. Specifically, if $g(x)=\sum_{j=0}^m a_jx^j$, then $g\in(x^2)$ if and only if $a_0=a_1=0$.
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It's all the polynomials in $\Bbb Z[x]$ with zero constant and $x$term.
$x^{2}$ is not a polynomial, so not an element of $\Bbb Z[x]$.