Editing Integral domain (section)

== Definition ==
An ''integral domain'' is a [[zero ring|nonzero]] [[commutative ring]] in which the product of any two nonzero elements is nonzero.  Equivalently:
* An integral domain is a nonzero commutative ring with no nonzero [[zero divisor]]s.
* An integral domain is a commutative ring in which the [[zero ideal]] {0} is a [[prime ideal]].
* An integral domain is a nonzero commutative ring for which every nonzero element is [[cancellation property|cancellable]] under multiplication.
* An integral domain is a ring for which the set of nonzero elements is a commutative [[monoid]] under multiplication (because a monoid must be [[closure (mathematics)| closed]] under multiplication).
* An integral domain is a nonzero commutative ring in which for every nonzero element ''r'', the function that maps each element ''x'' of the ring to the product ''xr'' is [[injective]]. Elements ''r'' with this property are called ''regular'', so it is equivalent to require that every nonzero element of the ring be regular.
* An integral domain is a ring that is [[isomorphic]] to a [[subring]] of a [[field (mathematics)|field]].  (Given an integral domain, one can embed it in its [[field of fractions]].)