subdirect product of rings

A ring R is said to be (represented as) a subdirect productPlanetmathPlanetmath of a family of rings {Ri:iI} if:

  1. 1.

    there is a monomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ε:RRi, and

  2. 2.

    given 1., πiε:RRi is surjectivePlanetmathPlanetmath for each iI, where πi:RiRi is the canonical projection map.

A subdirect product () of R is said to be trivial if one of the πiε:RRi is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

Direct productsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and direct sumsMathworldPlanetmathPlanetmath 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 /(pini).

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 commutativePlanetmathPlanetmathPlanetmath reduced ring is a field, a Boolean ringMathworldPlanetmath B can be represented as a subdirect product of 2. Furthermore, if this Boolean ring B is finite, the subdirect product becomes a direct product . Consequently, B has 2n elements, where n is the number of copies of 2.

Title subdirect product of rings
Canonical name SubdirectProductOfRings
Date of creation 2013-03-22 14:19:11
Last modified on 2013-03-22 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