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^{} $\epsilon :R\u27f6\prod {R}_{i}$, and

2.
given 1., ${\pi}_{i}\circ \epsilon :R\u27f6{R}_{i}$ is surjective^{} for each $i\in I$, where ${\pi}_{i}:\prod {R}_{i}\u27f6{R}_{i}$ is the canonical projection map.
A subdirect product () of $R$ is said to be trivial if one of the ${\pi}_{i}\circ \epsilon :R\u27f6{R}_{i}$ is an isomorphism^{}.
Direct products^{} and direct sums^{} of rings are all examples of subdirect products of rings. $\mathbb{Z}$ does not have nontrivial direct product nor nontrivial direct sum of rings. However, $\mathbb{Z}$ can be represented as a nontrivial 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 ${\mathbb{Z}}_{2}$.
Title  subdirect product of rings 

Canonical name  SubdirectProductOfRings 
Date of creation  20130322 14:19:11 
Last modified on  20130322 14:19:11 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  15 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 16D70 
Classification  msc 16S60 
Synonym  subdirect sum 
Defines  trivial subdirect product 