Editing Domain (ring theory) (section)

{{Short description|Unital ring with no zero divisors other than 0; noncommutative generalization of integral domains}}
In [[algebra]], a '''domain''' is a [[zero ring|nonzero]] [[ring (mathematics)|ring]] in which {{nowrap|1=''ab'' = 0}} implies {{nowrap|1=''a'' = 0}} or {{nowrap|1=''b'' = 0}}.<ref name="Lam">Lam (2001), p. 3</ref> (Sometimes such a ring is said to "have the [[zero-product property]]".) Equivalently, a domain is a ring in which 0 is the only left [[zero divisor]] (or equivalently, the only right zero divisor).   A [[commutative ring|commutative]] domain is called an [[integral domain]].<ref name="Lam" /><ref>Rowen (1994), p. 99.</ref>  Mathematical literature contains multiple variants of the definition of "domain".<ref>Some authors also consider the [[zero ring]] to be a domain: see Polcino M. & Sehgal (2002), p. 65.  Some authors apply the term "domain" also to [[rng (mathematics)|rngs]] with the zero-product property; such authors consider ''n'''''Z''' to be a domain for each positive integer ''n'': see Lanski (2005), p. 343.  But integral domains are always required to be nonzero and to have a 1.</ref>

{{Algebraic structures |Ring}}