Dictionary

product ring en · NOUN

Meanings

1. (qualifier:ring theory) A ring that is the direct product of rings.
   • For instance, let X be an infinite set, R the product ring K^(XY)#92;#33;, and A the set of homomorphisms R#92;rightarrowK given by evaluation at all the elements of X. Then the hypothesis of the Lemma holds, but #92;text#123;Int#125;#95;A(R) is the product ring k^(XY)#92;#33;, which is not in general free as a k-module. 1996, George M. Bergman, Adam O. Hausknecht, Cogroups and Co-rings in Categories of Associative Rings, American Mathematical Society, page 246:
   • Thus the set X#92;timesY becomes a ring with these operations and it is called the product ring. The identity element of the product ring with respect to the addition is obviously (0,0) where 0 and 0 are identity elements of addition in the rings X and Y#92;#33;, respectively. 2003, Erdoğan S. Şuhubi, Functional Analysis, Springer, page 63:
   • 2007, Catriona Maclean (translator), Daniel Perrin, Algebraic Geometry: An Introduction, [1995, D. Perrin, Géométrie algébrique], Springer, page 101, Then A_λ is isomorphic to the product ring k×k via the homomorphism sending X to (α,-α).

Forms

Deriveds

cup product ring · semidirect product ring

Synonyms

direct product ring (ring that is the direct product of rings)

Translations (2)

fr produit d'anneaux (ring that is the direct product of rings) · pl iloczyn pierścieni (ring that is the direct product of rings)