What is the group of units of Z, Q, R[x]???
I do not know where to start... any hints or clues...
No, actually a polynomial $\displaystyle f \in R[X]$ is not invertible if $\displaystyle \mathrm{deg}(f) \geq 1$. You need to show this. But then you're left with units of $\displaystyle R\subset R[X]$, the invertible polynomials of degree zero, wich is $\displaystyle R^*$
So, what is the group units of R?
Yes since determinant not 0 means it is invertible. I'm not sure about the purpose... but it can be useful for finding the zero divisors. If you have the list of all units, every nonzero element not in the list is a zero divisor. Because the zero divisors are precisely the nonzero nonunits.
This is maths! There doesn't have to be a purpose!
The group of units, however, is interesting. It can be used, for example, to find out if two rings are isomorphic or not (none of the three rings you gave are isomorphic as they have non-isomorphic groups of units).
What is also interesting is this:
A group is called Linear if it can be found as a subgroup of some general linear group. For example, all finite groups are linear groups. However, not all groups are linear groups! Now, there are rings called PI-rings, and if $\displaystyle G$ is a linear group then $\displaystyle G$ occurs as a group of units of a PI-rings (it is called a PI-group). Not all PI-groups are Linear groups.
That is, $\displaystyle \textnormal{Linear group }\subsetneq\textnormal{ PI-group}$.