Editing Commutative ring (section)

=== Definition ===
{{details|topic=the definition of rings|Ring (mathematics)}}
A ''ring'' is a [[Set (mathematics)|set]] <math> R </math> equipped with two [[binary operation]]s, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "<math>+</math>" and "<math>\cdot</math>"; e.g. <math>a+b</math> and <math>a \cdot b</math>. To form a ring these two operations have to satisfy a number of properties: the ring has to be an [[abelian group]] under addition as well as a [[monoid]] under multiplication, where multiplication [[distributive law|distributes]] over addition; i.e., <math>a \cdot \left(b + c\right) = \left(a \cdot b\right) + \left(a \cdot c\right)</math>. The identity elements for addition and multiplication are denoted <math> 0 </math> and <math> 1 </math>, respectively.

If the multiplication is commutative, i.e.
<math display="block">a \cdot b = b \cdot a,</math>
then the ring <math> R </math> is called ''commutative''. In the remainder of this article, all rings will be commutative, unless explicitly stated otherwise.