A ring $R$ is said to be (represented as) a subdirect product of a family of rings $\lbrace R_i: i \in I \rbrace$ if:

1. there is a monomorphism $\varepsilon: R \longrightarrow \prod R_i$ and
2. given 1., $\pi_i \circ \varepsilon: R \longrightarrow R_i$ is surjective for each $i \in I$ where $\pi_i:\prod R_i \longrightarrow R_i$ is the canonical projection map.

A subdirect product (representation) of $R$ is said to be trivial if one of the $\pi_i \circ \varepsilon: R \longrightarrow R_i$ is an isomorphism.

Direct products and direct sums of rings are all examples of subdirect products of rings. $\mathbb{Z}$ does not have non-trivial direct product nor non-trivial direct sum representations of rings. However, $\mathbb{Z}$ can be represented as a non-trivial subdirect product of $\mathbb{Z}/({p_i}^{n_i})$

As an application of subdirect products, it can be shown that any ring can be represented as a subdirect product of subdirectly irreducible rings. Since a subdirectly irreducible commutative reduced ring is a field, a Boolean ring $B$ can be represented as a subdirect product of $\mathbb{Z}_2$ Furthermore, if this Boolean ring $B$ is finite, the subdirect product representation becomes a direct product representation. Consequently, $B$ has $2^n$ elements, where $n$ is the number of copies of $\mathbb{Z}_2$