Subdirect product

In mathematics, especially in the areas of abstract algebra known as universal algebra, group theory, ring theory, and module theory, a subdirect product is a subalgebra of a direct product that depends fully on all its factors without however necessarily being the whole direct product. The notion was introduced by Birkhoff in 1944 and has proved to be a powerful generalization of the notion of direct product.
Definition.
A subdirect product is a subalgebra (in the sense of universal algebra) "A" of a direct product Π"iAi" such that every induced projection (the composite "pjs": "A" → "Aj" of a projection "p""j": Π"iAi" → "Aj" with the subalgebra inclusion "s": "A" → Π"iAi") is surjective.
A direct (subdirect) representation of an algebra "A" is a direct (subdirect) product isomorphic to "A".
An algebra is called subdirectly irreducible if it is not subdirectly representable by "simpler" algebras. Subdirect irreducibles are to subdirect product of algebras roughly as primes are to multiplication of integers.