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.

there is a monomorphism $\varepsilon:R\longrightarrow\prod R_{i}$, and

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.

A subdirect product () 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 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 $\mathbb{Z}_{2}$.

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