Determine the units of the ring $A= \mathbb Z[X]/(X^3)$ and the structure of the group $A^*$ Determine the units of the ring $A= \mathbb Z[X]/(X^3)$ and the structure of the group $A^*$
I've only managed to show that the free coefficient of any unit in $A$ is a unit in $\mathbb Z$.

Hint: as you noted, a necessary condition is that the constant term be equal to $\pm1$. Conversely, if $P$ has constant term equal to $\pm 1$, then $\pm P=1+XQ$ is a unit, since it is the sum of a unit and a nilpotent element.
As for the group structure, you have a bijection
$$\phi:\lbrace -1,+1\rbrace\times\Bbb Z^2\to A^\times, (\epsilon,a,b)\mapsto\epsilon+aX+bX^2$$
and the multiplicative structure is given by
$$\begin{eqnarray}\phi(\epsilon,a,b)\times\phi(\epsilon',a',b')&=&\epsilon\epsilon'+(a\epsilon'+a'\epsilon)X+(b\epsilon'+b'\epsilon+aa')X\\&=&\phi(\epsilon\epsilon',a\epsilon'+a'\epsilon,b\epsilon'+b'\epsilon+aa')\end{eqnarray}$$
which describes the group law.
You can interpret this matricially:
$$\begin{pmatrix}\epsilon&a&b\\0&\epsilon&a\\0&0&\epsilon\end{pmatrix}
\times
\begin{pmatrix}\epsilon'&a'&b'\\0&\epsilon'&a'\\0&0&\epsilon'\end{pmatrix}=
\begin{pmatrix}\epsilon\epsilon'&a\epsilon'+a'\epsilon&b\epsilon'+b'\epsilon+aa'\\0&\epsilon\epsilon'&a\epsilon'+a'\epsilon\\0&0&\epsilon\epsilon'\end{pmatrix}$$
So $A^\times$ identifies with a commutative subgroup of $\mathrm{GL}_2(\Bbb Z)$.
