Problem: Find the zero divisors and the units of the quotient ring $\mathbb Z[X]/\langle X^3 \rangle$.
If $a \in \mathbb Z[X]/ \langle X^3 \rangle$ is a zero divisor, then there is $b \neq 0_I$ such that $ab=0_I$. I think that the elements $a=X+ \langle X^3 \rangle$ and $b=X^2+ \langle X^3 \rangle$ are zero divisors because we have:
$$ab=XX^2+ \langle X^3 \rangle =X^3+ \langle X^3 \rangle = \langle X^3 \rangle.$$
I couldn't think of any other divisors so I suspect these two are the only ones. Am I correct? If that is the case, how could I show these are the only zero divisors?
As for the units I don't know what to do. Any suggestions would be appreciated.