Not to be confused with domain of integration. In mathematics, and specifically abstract algebra an introduction hungerford pdf abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. 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.
Some sources, notably Lang, use the term entire ring for integral domain. An integral domain is basically defined as a nonzero commutative ring in which the product of any two nonzero elements is nonzero. An integral domain is a nonzero commutative ring with no nonzero zero divisors. An integral domain is a nonzero commutative ring for which every non-zero element is cancellable 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. A fundamental property of integral domains is that every subring of a field is an integral domain, and that, conversely, given any integral domain, one may construct a field that contains it as a subring, the field of fractions.
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Every field is an integral domain. Conversely, every Artinian integral domain is a field. Rings of polynomials are integral domains if the coefficients come from an integral domain. The previous example can be further exploited by taking quotients from prime ideals. The same is true for rings of analytic functions on connected open subsets of analytic manifolds. A regular local ring is an integral domain.