What are the elements of the quotient ring $\mathbb{Z_3}[x]/(x^3 + x^2)$? $$R = \mathbb{Z_3}[x]/(x^3 + x^2).$$
As $\mathbb{Z_3}$ is a field we have that every polynomial in $\mathbb{Z_3}[x]/(x^3 + x^2)$ of degree less than ${x^3 + x^2}$ is a distinct element in $R$.
So I conclude the following are the elements of $R$.
$
\overline{0},
\overline{1},
\overline{2}$
$
\overline{x},
\overline{x + 1},
\overline{x + 2},
\overline{2x},
\overline{2x + 1},
\overline{2x + 2},
$
$
\overline{x^2},
\overline{x^2 + 1},
\overline{x^2 + 2},
\overline{x^2 +x},
\overline{x^2 + 2x},
\overline{x^2 + x + 1},
\overline{x^2 + x + 2},
\overline{x^2 + 2x + 1},
\overline{x^2 + 2x + 2},
$
$
\overline{2x^2},
\overline{2x^2 + 1},
\overline{2x^2 + 2},
\overline{2x^2 +x},
\overline{2x^2 + 2x},
\overline{2x^2 + x + 1},
\overline{2x^2 + x + 2},
\overline{2x^2 + 2x + 1},
\overline{2x^2 + 2x + 2},$
Also, noting that for each element in $R$ we have $3$ choices for the coefficient and $3$ choices for the exponent of $x \implies$ we have $3^3 = 27$ polynomials in $R$.
Have I got my facts right here?

As pointed out in the comments, you are correct. The ring $\mathbb{Z}_3[x]/(x^3 + x^2)$ has $27$ elements, and they are precisely the ones you wrote down.
