A functor from semigroups to rings

The functor Z in question associates to every semigroup S the semigroup ring Z(S) where Z is the ring of integers. We construct two more functors, one naturally equivalent to Z and one adjoint of Z. In relation to the functor Z, we also study the functor which to each ring R associates the usual embedding of R into a unitary ring. We consider the effect of the functor Z on such semigroup constructions as strong semilattices of semigroups, Rees matrix and Bruck semigroups. The paper concludes with certain commutativity property of Z with the translational hull in semigroups and rings.