subdirect product of rings
A ring is said to be (represented as) a subdirect product of a family of rings if:
there is a monomorphism , and
Direct products and direct sums of rings are all examples of subdirect products of rings. does not have non-trivial direct product nor non-trivial direct sum of rings. However, can be represented as a non-trivial subdirect product of .
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 can be represented as a subdirect product of . Furthermore, if this Boolean ring is finite, the subdirect product becomes a direct product . Consequently, has elements, where is the number of copies of .
|Title||subdirect product of rings|
|Date of creation||2013-03-22 14:19:11|
|Last modified on||2013-03-22 14:19:11|
|Last modified by||CWoo (3771)|
|Defines||trivial subdirect product|