Characterization of direct product of rings

Question

Characterization of the direct product of rings

I'm trying to understand the given characterization of direct products of rings. More specifically, I realize that given a direct product the statement holds, but how exactly does it characterize the direct product?

Answer 1

If there is any other ring $S$ satisfying the characterization, then $S$ is isomorphic to $\prod_{i=1}^n R_i$ via unique isomorphism. The proof is standard and works for any category where direct products exist.

These kind of characterizations are usually called universal properties.

Answer 2

It characterizes the direct product in the sense that every other ring with that property is isomorphic to the direct product.

That is, the direct product has a universal property that makes it unique (up to isomorphism.)