If $AB = I$ then $BA = I$
Most introductory linear algebra texts define the inverse of a square matrix $A$ as such:
Inverse of $A$, if it exists, is a matrix $B$ such that $AB=BA=I$.
That definition, in my opinion, is problematic. A few books (in my sample less than 20%) give a different definition:
Inverse of $A$, if it exists, is a matrix $B$ such that $AB=I$. Then they go and prove that $BA=I$.
Do you know of a proof other than defining inverse through determinants or through using
Is there a general setting in algebra under which $ab=e$ leads to $ba=e$ where $e$ is the identity?