product ring

product ring (plural product rings)

1. (mathematics, ring theory) A ring that is the direct product of rings.
   • 1996, George M. Bergman, Adam O. Hausknecht, Cogroups and Co-rings in Categories of Associative Rings, American Mathematical Society, page 246,

     For instance, let ${\displaystyle X}$ be an infinite set, ${\displaystyle R}$ the product ring ${\displaystyle K^{X}Y\!,}$ and A the set of homomorphisms ${\displaystyle R\rightarrow K}$ given by evaluation at all the elements of ${\displaystyle X}$ . Then the hypothesis of the Lemma holds, but ${\displaystyle {\text{Int}}_{A}(R)}$ is the product ring ${\displaystyle k^{X}Y\!,}$ which is not in general free as a k-module.

   • 2003, Erdoğan S. Şuhubi, Functional Analysis, Springer, page 63,

     Thus the set ${\displaystyle X\times Y}$ 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 ${\displaystyle (0,0)}$ where 0 and 0 are identity elements of addition in the rings ${\displaystyle X}$ and ${\displaystyle Y\!,}$ respectively.

   • 2007, Catriona Maclean (translator), Daniel Perrin, Algebraic Geometry: An Introduction, [1995, D. Perrin, Géométrie algébrique], Springer, page 101,

     Then ${\displaystyle A_{\lambda }}$ is isomorphic to the product ring ${\displaystyle k\times k}$ via the homomorphism sending ${\displaystyle X}$ to ${\displaystyle (\alpha ,-\alpha )}$ .

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

• cup product ring
• semidirect product ring