Proof that set of units of a ring is a multiplicative group

Question

How can I proof that the set of units of a ring is a multiplicative group?

If I look at $\mathbb{Z}$ I have $\mathbb{Z}^*=\{-1,1\}$

Is it sufficient to say that $\mathbb{Z}^*$ is a subset of $\mathbb{Z}$ and since $\mathbb{Z}$ is a ring (which implies that it is also a group) its subset is also a group?

Or more general (for all sets): If $R$ is a ring, $R$ is also a group and $R^*$ is a subset of $R$ and thus a multiplicative (sub-)group.

Answer 1

A group is a set with an associative operation that has identity and inverse.

The ring structure grants you the associative law and the identity (wich is obviously an unit), so you only have to prove two facts:

1. The set of units is closed under product, that is, the product of two units is also an unit.
2. Every unit has an inverse and this inverse is also an unit.