Enlargements and Morita contexts for semirings

In this paper we study enlargements and Morita contexts for factorisable semirings. These are semirings where each element is a finite sum of products of some elements. We show that a full matrix semiring over a factorisable semiring is an enlargement of that semiring. We call two factorisable semirings Morita equivalent if they are connected by a unitary and surjective Morita context. It turns out that two factorisable semirings are Morita equivalent if and only if they have a joint enlargement. We prove that each unitary Morita context between factorisable semirings is isomorphic to a Morita context that is induced by a joint enlargement of those semirings. Finally we study Morita invariants for semirings. Differently from the case of rings or semigroups, regularity is not a Morita invariant for semirings with identity. We prove that if two commutative factorisable nondegenerate semirings are Morita equivalent, then they are isomorphic.