product ring

English

Noun

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

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

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

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

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

Synonyms

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

Derived terms

• cup product ring
• semidirect product ring

Translations

ring that is the direct product of rings

• French: produit d’anneaux m
• (deprecated template usage) {{trans-mid}}