In this article Wikipedia defines invertibility for square matrices over commutative rings as follows:
...in the case of the ring being commutative, the condition for a square matrix to be invertible is that its determinant is invertible in the ring
But for non-commutative rings, it does not offer a definition:
For a noncommutative ring, the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings.
I was wondering what that definition is. Can anyone help me find it?