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 $\\varepsilon:R\\longrightarrow\\prod R_{i}$ , 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 () 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 $\\mathbb{Z}_{2}$ .

单词	SubdirectProductOfRings
释义	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 $\\varepsilon:R\\longrightarrow\\prod R_{i}$ , 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 () 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 $\\mathbb{Z}_{2}$ .
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