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.[citation needed]
group theory, ring theory, and module theory, a subdirectproduct is a subalgebra of a direct product that depends fully on all its factors without however...
lemma, every subdirectproduct is a fiber product. Let G, H, and Q be groups, and let 𝜑: G → Q and χ: H → Q be homomorphisms. The fiber product of G and...
applications), a subdirectly irreducible algebra is an algebra that cannot be factored as a subdirectproduct of "simpler" algebras. Subdirectly irreducible...
only if it is a subdirectproduct of left primitive rings. A commutative ring is semiprimitive if and only if it is a subdirectproduct of fields, (Lam...
is subdirectly irreducible; when R is written as a subdirectproduct of rings, then one of the projections of R onto a ring in the subdirectproduct is...
that every distributive lattice is a subdirectproduct of copies of the two-element chain, or that the only subdirectly irreducible member of the class of...
rings are unit regular. Every strongly von Neumann regular ring is a subdirectproduct of division rings. In some sense, this more closely mimics the properties...
just a semisimple ring. Semiprimitive rings can be understood as subdirectproducts of primitive rings, which are described by the Jacobson density theorem...
and 0 {\displaystyle 0} otherwise. Every skew Boolean algebra is a subdirectproduct of primitive algebras. Skew Boolean algebras play an important role...
{2}})} ). More generally the real closure of a field F is a certain subdirectproduct of the real closures of the ordered fields (F,P), where P runs through...
ISBN 0-521-78451-4, 10.23 Infinite distributive laws, pp. 239–240 G. N. Raney, A subdirect-union representation for completely distributive complete lattices, Proceedings...
(and forms another Heyting algebra) is subdirectly irreducible, whence every Heyting algebra can be made subdirectly irreducible by adjoining a new greatest...
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