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If A, B are integral domains over an algebraically closed field k, then A ⊗ k B is an integral domain. This is a consequence of Hilbert's nullstellensatz , [ a ] and, in algebraic geometry, it implies the statement that the coordinate ring of the product of two affine algebraic varieties over an algebraically closed field is again an integral ...
In particular, this means that any element of L integral over A is root of a monic polynomial in A[X] that is irreducible in K[X]. If A is a domain contained in a field K, we can consider the integral closure of A in K (i.e. the set of all elements of K that are integral over A). This integral closure is an integrally closed domain.
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.
If R is an integral domain, then R[t] is also an integral domain; its field of fractions is the field of rational functions. If R is a Noetherian ring, then R[t] is a Noetherian ring. If R is a unique factorization domain, then R[t] is a unique factorization domain. Finally, R is a field if and only if R[t] is a principal ideal domain.
An integral domain is called a Dedekind domain if for each pair of ideals , there is an ideal such that = . [18] It can then be shown that every nonzero ideal of a Dedekind domain can be uniquely written as a product of maximal ideals, a generalization of the fundamental theorem of arithmetic .
The field of fractions of an integral domain is sometimes denoted by or (), and the construction is sometimes also called the fraction field, field of quotients, or quotient field of . All four are in common usage, but are not to be confused with the quotient of a ring by an ideal , which is a quite different concept.
With those tools, the Leibniz integral rule in n dimensions is [4] = () + + ˙, where Ω(t) is a time-varying domain of integration, ω is a p-form, = is the vector field of the velocity, denotes the interior product with , d x ω is the exterior derivative of ω with respect to the space variables only and ˙ is the time derivative of ω.
An integral domain is a UFD if and only if it is a GCD domain (i.e., a domain where every two elements have a greatest common divisor) satisfying the ascending chain condition on principal ideals. An integral domain is a Bézout domain if and only if any two elements in it have a gcd that is a linear combination of the two.