Multiplicative Group of a Ring

Proof

This proof is fairly simple, and really comes down to proving the a product of units is a unit.

First note that $1\in R$ is a unit, since $1\cdot 1=1⟹1=1^{-1}$.

Now, for $a,b\in R^{\ast }$ we have that:

hence $(ab{)}^1$ exists and is equal to $(b^{-1}{a}^{-1}$ (note that showing it is a left inverse is basically identical to the above).

Now, $a^{-1}\in R^{\ast }$ since $a^{-1}a=1⟹(a^{-1}{)}^{-1}=a$.

Finally, associativity is inherited from the properties of $R$.