Inversion of matrices over a commutative semiring (Q1057341)

A well known result in elementary matrix theory is that for square matrices, A, B, \(AB=I\) implies \(BA=I\). This result is valid for \(n\times n\) matrices over a commutative ring as well as over a field. It is not true in general, however, for noncommutative rings. The main result of the present paper is the extension of this result to commutative semirings. Specifically the authors prove: If R is a commutative semiring and A,B are \(n\times n\) matrices over R then \(AB=I\) implies \(BA=I\). The authors also discuss a related property, called stability and show, with a suitable example, that the semiring \(R_ n\) of \(n\times n\) matrices over a commutative semiring R is not necessarily stable.