# Math Help - Units in a matrix ring

1. ## Units in a matrix ring

Let $R$ be a commutative ring with 1. Characterize the units of $M_n(R)$.

If $R$ is a field, then the usual characterization of the units are those matrices with non-zero determinant. But since $R$ is just a commutative ring with 1 (and not even an integral domain) I have no idea how to characterization the units. Is there any nice way to do this?

2. ## Re: Units in a matrix ring

Originally Posted by jgensler1992
Let $R$ be a commutative ring with 1. Characterize the units of $M_n(R)$.
$A$ is a unit of $\textrm{M}_n(R)$ if and only if $|A|$ a is a unit of $R$ .

Hint : Use $(\textrm{adj}\;B)B=B(\textrm{adj}\;B)=|B|I_n$ .