# Find the zero divisors and the units of $\mathbb Z[X]/<X^3>$

1. Sep 13, 2014

### mahler1

The problem statement, all variables and given/known data

Find the zero divisors and the units of the quotient ring $\mathbb Z[X]/<X^3>$

The attempt at a solution

If $a \in \mathbb Z[X]/<X^3>$ is a zero divisor, then there is $b \neq 0_I$ such that $ab=0_I$. I think that the elements $a=X+<X^3>$ and $b=X^2+<X^3>$ are zero divisors because we have

$ab=XX^2+<X^3>=X^3+<X^3>=<X^3>$.

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.

2. Sep 18, 2014