subdirect product of rings

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

1. there is a monomorphism $\epsilon :R⟶\prod R_i$, and

2. given 1., ${\pi }_i\circ \epsilon :R⟶R_i$ is surjective for each $i\in I$, where ${\pi }_i:\prod R_i⟶R_i$ is the canonical projection map.

A subdirect product () of $R$ is said to be trivial if one of the ${\pi }_i\circ \epsilon :R⟶R_i$ 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 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 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 becomes a direct product . Consequently, $B$ has $2^n$ elements, where $n$ is the number of copies of ${\mathbb{Z}}_2$.