Meaning of product ring | Babel Free

Definitions

A ring that is the direct product of rings.

Equivalents

Examples

“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.”

“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.”

“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).”

CEFR level