Editing Commutative ring (section)

=== First examples ===
An important example, and in some sense crucial, is the [[integer|ring of integer]]s <math> \mathbb{Z} </math> with the two operations of addition and multiplication. As the multiplication of integers is a commutative operation, this is a commutative ring. It is usually denoted <math> \mathbb{Z} </math> as an abbreviation of the [[German language|German]] word ''Zahlen'' (numbers).

A [[field (mathematics)|field]] is a commutative ring where <math> 0 \neq 1 </math> and every [[0 (number)|non-zero]] element <math> a </math> is invertible; i.e., has a multiplicative inverse <math> b </math> such that <math> a \cdot b = 1 </math>. Therefore, by definition, any field is a commutative ring. The [[rational number|rational]], [[real number|real]] and [[complex number]]s form fields.

If <math> R </math> is a given commutative ring, then the set of all [[polynomial]]s in the variable <math> X </math> whose coefficients are in <math> R </math> forms the [[polynomial ring]], denoted <math> R \left[ X \right] </math>. The same holds true for several variables.

If <math> V </math> is some [[topological space]], for example a subset of some <math> \mathbb{R}^n </math>, real- or complex-valued [[continuous function]]s on <math> V </math> form a commutative ring. The same is true for [[differentiable function|differentiable]] or [[holomorphic function]]s, when the two concepts are defined, such as for <math> V </math> a [[complex manifold]].