Editing Direct product (section)

== Examples ==
* If <math>\R</math> is thought of as the set of [[real numbers]] without further structure, the direct product <math>\R \times \R</math> is just the Cartesian product <math>\{(x,y) : x,y \in \R\}.</math>
* If <math>\R</math> is thought of as the [[group (mathematics)|group]] of real numbers under addition, the direct product <math>\R\times \R</math> still has <math>\{(x,y) : x,y \in \R\}</math> as its underlying set. The difference between this and the preceding examples is that <math>\R \times \R</math> is now a group and so how to add their elements must also be stated. That is done by defining <math>(a,b) + (c,d) = (a+c, b+d).</math>
* If <math>\R</math> is thought of as the [[ring (mathematics)|ring]] of real numbers, the direct product <math>\R\times \R</math> again has <math>\{(x,y) : x,y \in \R\}</math> as its underlying set. The ring structure consists of addition defined by <math>(a,b) + (c,d) = (a+c, b+d)</math> and multiplication defined by <math>(a,b) (c,d) = (ac, bd).</math>
* Although the ring <math>\R</math> is a [[field (mathematics)|field]], <math>\R \times \R</math> is not because the nonzero element <math>(1,0)</math> does not have a [[multiplicative inverse]].

In a similar manner, the direct product of finitely many algebraic structures can be talked about; for example, <math>\R \times \R \times \R \times \R.</math> That relies on the direct product being [[associative]] [[up to]] [[isomorphism]]. That is, <math>(A \times B) \times C \cong A \times (B \times C)</math> for any algebraic structures <math>A,</math> <math>B,</math> and <math>C</math> of the same kind. The direct product is also [[commutative]] up to isomorphism; that is, <math>A \times B \cong B \times A</math> for any algebraic structures <math>A</math> and <math>B</math> of the same kind. Even the direct product of infinitely many algebraic structures can be talked about; for example, the direct product of [[countably infinite|countably]] many copies of <math>\mathbb R,</math> is written as <math>\R \times \R \times \R \times \dotsb.</math>