Editing Integral domain (section)

{{Short description|Commutative ring with no zero divisors other than zero}}
{{Use American English|date = March 2019}}
{{distinguish |text= [[ Integral | domain of integration]]}}
{{Ring theory sidebar}}
In [[mathematics]], an '''integral domain''' is a [[Zero ring|nonzero]] [[commutative ring]] in which [[zero-product property|the product of any two nonzero elements is nonzero]].{{sfn|Bourbaki|1998|p=116|ps=none}}{{sfn|Dummit|Foote|2004|p=228|ps=none}} Integral domains are generalizations of the [[ring (mathematics)|ring]] of [[integer]]s and provide a natural setting for studying [[divisibility (ring theory)|divisibility]].  In an integral domain, every nonzero element ''a'' has the [[cancellation property]], that is, if {{nowrap|''a'' ≠ 0}}, an equality {{nowrap|''ab'' {{=}} ''ac''}} implies {{nowrap|''b'' {{=}} ''c''}}.

"Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a [[multiplicative identity]], generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity.{{sfn|van der Waerden|1966|p=36|ps=none}}{{sfn|Herstein|1964|pp=88–90|ps=none}} Noncommutative integral domains are sometimes admitted.{{sfn|McConnell|Robson|ps=none}}  This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "[[domain (ring theory)|domain]]" for the general case including noncommutative rings.

Some sources, notably [[Serge Lang|Lang]], use the term '''entire ring''' for integral domain.{{sfn|Lang|1993|pp=91–92|ps=none}}

Some specific kinds of integral domains are given with the following chain of [[subclass (set theory)|class inclusions]]:

{{Commutative ring classes}}

{{Algebraic structures |Ring}}