$AB=I$ implies $BA=I$ in a $M_n(R)$ when $R$ is a semiring? Let $R$ be a unitary commutative semi-ring and let $M_n(R)$ be the semi-ring of $n \times n$ matrices with coefficients in $R$.  Let's call $I$ the identity matrix in $M_n(R)$ (it exists because $R$ is unitary).

Is it true that for any $A$ and $B$ in $M_n(R)$ such that $AB=I$ we have $BA=I$?

 A: This is proven in Inversion of Matrices over a Commutative Semiring by Reutenauer and Straubing.
The proofs aren't especially short though (as you request in the comments).
The first two paragraphs give good context for their paper though:

It is a well-known consequence of the elementary theory of vector spaces
  that if $A$ and $B$ are $n$-by-$n$ matrices over a field (or even a skew field) such
  that $AB = 1$, then $BA = 1$. This result remains true for matrices over a
  commutative ring, however, it is not, in general, true for matrices over noncommutative
  rings.
In this paper we show that if $A$ and $B$ are $n$-by-$n$ matrices over a
  commutative semiring, then the equation $AB = 1$ implies $BA = 1$. We give
  two proofs: one algebraic in nature, the other more combinatorial. Both
  proofs use a generalization of the familiar product law for determinants over
  a commutative semiring.

It's worth mentioning that "semi-ring" for them requires having a multiplicative identity, so they do prove the result you desire.
