Left Adjoint to the Forgetful Functor from Rings to Semirings.

Question

Is there a left adjoint to the forgetful functor from the category of commutative rings to the category of commutative semirings?

A semiring is like a ring, except for the existence of an additive inverse.

Answer 1

Given a commutative semiring $S$, we can define a ring structure on $S\times S/{\sim}$ where $(a,b)\sim(c,d)\leftrightarrow a+d=b+c$ by letting $[a,b]+[c,d]=[a+c,b+d]$ and $[a,b]\cdot[c,d]=[ac+bd,ad+bc]$.

If $S$ is a commutative semiring and $A$ a commutative ring, then a semiring homomorphism $f\colon S\to A$ determines a ring homomorphism $S\times S/{\sim}\to A$, $[a,b]\mapsto f(a)-f(b)$, and vice versa a ring homomorphism $g\colon S\times S/{\sim}\to A$ determines a semiring homomoprhism $S\to A$, $x\mapsto g([x,0])$. Clearly, these associations are inverses of each other, as desired.