Subdirect product

Remarks

Let $(\theta_i)_{i \in I}$ be a family of congruences on an algebra $\mathbf{A}$. The map $a \mapsto (a/\theta_i)_{i \in I}$ is a subdirect embedding of $\mathbf{A}$ into $\prod_{i \in I} \mathbf{A}/\theta_i$ if and only if $\bigcap_{i \in I} \theta_i = \Delta_A$.

Example: the map $n \mapsto (n \bmod p)_p$ is a subdirect embedding of the ring $\Z$ into $\prod_{p \text{ prime}} \Z/p\Z$. It is a homomorphism, it is injective because a nonzero integer has only finitely many prime divisors, and each coordinate map $\Z \to \Z/p\Z$ is onto.

Non-example: the subgroup $\set{(x, 0) : x \in \Z/2\Z}$ of $\Z/2\Z \times \Z/2\Z$ is not a subdirect product. Its projection onto the second factor is $\set{0}$, not all of $\Z/2\Z$.