TITLE: Units in quotient ring of $\mathbb Z[X]$
QUESTION [6 upvotes]: An exercise from Dummit & Foote: 

Determine the units of the ring $A = \mathbb{Z}[X]/(X^{3})$ and the structure of the unit group $A^{\times}$.

Help would be great.
Thanks!

REPLY [5 votes]: Actually nobody answered the second part of the problem about the structure of the unit group $A^{\times}.$ The answer is the following: 

$A^{\times}$ is isomorphic to the abelian group $\mathbb{Z}_2\times\mathbb{Z}^2$. 

Hint. The unit group can be written as a direct sum of the subgroups generated by $-1$, $1+x$, and $1+x^2$ respectively. The subgroup generated by $-1$ is the torsion subgroup of $A^{\times}$, whilst the subgroup generated by $1+x$ and $1+x^2$ is the free part of $A^{\times}$.
Of course, the problem (and its answer) can be easily generalized to the ring $\mathbb{Z}[X]/(X^n)$.