Condensation of semirings

Every semiring is canonically associated to a minimal congruence with naturally ordered quotient, which can be defined by the additive J classes of the semiring. The quotient by this minimal congruence is the condensation of $S$.

The condensation theory of semirings
All the basic ingredients in this proof are available in any basic text on semiring theory. The fundamental breakthrough here is just a result in a change in thinking related to semiring congruences.

Theorem. let $S$ be a semiring and let $H$ be the Green's $H$ relation of the additive monoid $+$ of $S$, then $H$ forms a semiring congruence of $S$.

Proof. let $a,b,c,d \in S$ with $a H b$ and $c H d$. Then by the definition of the Green's $H$ relation there exist elements $x_1,x_2,y_1,y_2$ such that $$a + x_1 = b$$ $$a = b + y_1$$ $$c + x_2 = d$$ $$c = d + y_2$$ We can now demonstrate that $(a + c) \space H \space (b + d)$ by a simple process of substitution: $$a + c = b + d + y+1 + y_2$$ $$b + d = a + c + x_1 + x_2$$ To demonstrate that $ac \space H \space bd$ we can simply use the distributive law once after substitution: $$ac = (b + y_1)(d+ y_2) = bd + by_2 + dy_1 + y_1y_2$$ $$bd = (a + x_1)(c + x_2) = ac + ax_2 + cx_1 + x_1x_2$$ The fact that $a H b$ and $c H d$ both imply that $a + c \space H \space b + d$ and $ac \space H \space bd$ means that $H$ is both an additive congruence and a multiplicative congruence. It follows that it is a semiring congruence. $\square$

Definition. let $S$ be a semiring and let $H$ be the $H$ classes of its additive monoid then the condensation of $S$ is the quotient $\frac{S}{H}$.

The additive monoid of $S$ is its condensation on the level of semigroup theory, so it is not hard to see that it is a $J$ trivial. This is equivalent to saying that $S$ is a naturall ordered semiring.

Corollary. $\frac{S}{H}$ is a naturally ordered semiring

This condensation mapping $\pi : S \to \frac{S}{H}$ is characterized by a universal property in the sense of category theory.

Corollary. let $S$ be a semiring then any mapping $f$ from $S$ to a naturally ordered semiring $R$ filters through the condensation homomorphism $\pi : S \to \frac{S}{H}$ by a unique morphism $m$. General structure theory of semirings:
This condensation theorem for semirings is the most general tool we have for defining a general structure theory for semirings. This is applicable to any semiring, and it characterizes the relationship between its order theoretic and algebraic properties.

* Every semiring is an extension of a partially ordered semiring.

The only case when the partial order doesn't matter is the case in which is trivial, which is rings. For a ring $R$ it is clearly the case that all elements are additively related, so its quotient is the trivial semiring. Every semiring has a maximal subring, which is generated by the set of all elements with additive inverses.

References:
semiring in nLab