Нөлдік бөлгіштері жоқ сақиналар және домендердің жалпылама емес түрі

Unital ring with no zero divisors other than 0; noncommutative generalization of integral domains
In algebra, a domain is a nonzero ring in which 1=ab = 0 implies 1=a = 0 or 1=b = 0. (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. Mathematical literature contains multiple variants of the definition of "domain".

The ring is not a domain, because the images of 2 and 3 in this ring are nonzero elements with product 0. More generally, for a positive integer , the ring is a domain if and only if is prime. A finite domain is automatically a finite field, by Wedderburn's little theorem. The quaternions form a noncommutative domain. More generally, any division algebra is a domain, since every nonzero element is invertible. The set of all Lipschitz quaternions, that is, quaternions of the form where a, b, c, d are integers, is a noncommutative subring of the quaternions, hence a noncommutative domain. Similarly, the set of all Hurwitz quaternions, that is, quaternions of the form where a, b, c, d are either all integers or all half integers, is a noncommutative domain. A matrix ring Mn(R) for n ≥ 2 is never a domain: if R is nonzero, such a matrix ring has nonzero zero divisors and even nilpotent elements other than 0. For example, the square of the matrix unit E12 is 0. The tensor algebra of a vector space, or equivalently, the algebra of polynomials in noncommuting variables over a field, is a domain. This may be proved using an ordering on the noncommutative monomials. If R is a domain and S is an Ore extension of R then S is a domain. The Weyl algebra is a noncommutative domain. The universal enveloping algebra of any Lie algebra over a field is a domain. The proof uses the standard filtration on the universal enveloping algebra and the Poincaré–Birkhoff–Witt theorem.

Zero divisors have a topological interpretation, at least in the case of commutative rings: a ring R is an integral domain if and only if it is reduced and its spectrum Spec R is an irreducible topological space. The first property is often considered to encode some infinitesimal information, whereas the second one is more geometric. An example: the ring k[x, y]/(xy), where k is a field, is not a domain, since the images of x and y in this ring are zero divisors. Geometrically, this corresponds to the fact that the spectrum of this ring, which is the union of the lines 1=x = 0 and 1=y = 0, is not irreducible. Indeed, these two lines are its irreducible components.