TITLE: Ways to formalise $\text{Ring}\approx \text{Group}\times \text{Monoid}$.
QUESTION [6 upvotes]: In a (unit) ring, elements (of a set $S$) are able to operate on each other via $\cdot,+$. If we are to consider the maps $M:S\times S\to S:(a,b)\mapsto a+b$ and $G:S \times S \to S: (a,b )\mapsto a\cdot b$, we have that $\left \langle S,M \right \rangle$ is a monoid and $\left \langle S,G \right \rangle$ is a group. If we define the composition of maps in $G$ and $M$ such that they are distributive, we have a unit ring.
It is clear that many abstract-algebraic objects (fields, commutative rings, etc.) can arise from taking the 'product' of two other objects in a method similar to above. 
Is there a category-theoretical way to view the 'product' of two abstract-algebraic objects in a general sense? Are there any applications of this?

REPLY [6 votes]: The monad of rings is the composite of a distributive law of the monad for monoids over the monad for abelian groups. Cf. Jon Beck, "Distributive laws", Lecture Notes in Mathematics 80, p. 119-140, 1969.