# How is invertibility defined for square matrices over non-commutative rings?

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?

• For commutative rings, there is an easy criterion between invertibility and the determinant. For noncommutative rings the same criterion is not available. It does not mean the definition of invertible is different. Nov 18, 2022 at 13:25

It seems like they (justly) don't define invertible matrices by the determinant, but instead by the relation $$AB = BA = I$$, which is the good notion of invertibility when looking at matrices as a ring.