Subdirect Products, An Introduction

Introduction

A subdirect product is an injective ring homomorphism h from a ring R into a direct product of rings S₁*S₂*S₃*…, such that h is surjective on each S_i. (Compose h with the projection onto S_i and get all of S_i.) There is no need for S_i to equal S_j.

Note that R is a subring of the product ring. Since every ring homomoorphism carries 1 to 1, project down to S_i, and h maps 1 in R to 1 in S_i.

Unlike direct product, there are many nonisomorphic subdirect products of S_i. We'll see this later on.

Subdirectly Irreducible

The representation of R is trivial if some h_i is an isomorphism. In other words, R = S_i. The other components are merely images of R that go along for the ride.

To avoid a trivial representation, each h_i must have a nonzero kernel, and since R embeds, the intersection of these kernels K_i is 0.

If R has only trivial subdirect product representations, R is subdirectly irreducible.