# 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. 1.
2. 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.

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 $\mathbb{Z}_{2}$.

Title subdirect product of rings SubdirectProductOfRings 2013-03-22 14:19:11 2013-03-22 14:19:11 CWoo (3771) CWoo (3771) 15 CWoo (3771) Definition msc 16D70 msc 16S60 subdirect sum trivial subdirect product