Domain (ring theory)

In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0.^[1] (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 domain is called an integral domain.^[1]^[2] Mathematical literature contains multiple variants of the definition of "domain".^[3]

^ ^a ^b Lam (2001), p. 3
^ Rowen (1994), p. 99.
^ 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 rngs with the zero-product property; such authors consider nZ 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.

[Lam-1] Lam (2001), p. 3

[2] Rowen (1994), p. 99.

[3] 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 rngs with the zero-product property; such authors consider nZ 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.

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