product ring
noun1
Mathematics, Sciences 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:
1 more example
- 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).
- Synonyms
- direct product ring
- Derived terms
- cup product ring, semidirect product ring