If a ring R is commutative, I don't understand why if A,B∈Rn×n, AB=1 means that BA=1, i.e., Rn×n is Dedekind finite.
Arguing with determinant seems to be wrong, although det(AB)=det(BA)=1 but it necessarily doesn't mean that BA=1.
And is every left zero divisor also a right divisor ?
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