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An inductive limit of integral domains is an integral domain. 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 ...
This integral closure is an integrally closed domain. Integrally closed domains also play a role in the hypothesis of the Going-down theorem. The theorem states that if A⊆B is an integral extension of domains and A is an integrally closed domain, then the going-down property holds for the extension A⊆B.
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.
In mathematics, an integral is the ... if the values of a function are rearranged over the domain, the integral of a function should remain the same. ... The discrete ...
Discrete mathematics is the study of mathematical structures ... (if its domain is ... where there are integral transforms in harmonic analysis for studying ...
For example, any principal ideal domain is a unique factorization domain (UFD) which means that any element is a product of irreducible elements, in a (up to reordering of factors) unique way. Here, an element a {\displaystyle a} in a domain is called irreducible if the only way of expressing it as a product a = b c , {\displaystyle a=bc,} is ...
Discrete integral calculus is the study of the definitions, properties, and applications of the Riemann sums. The process of finding the value of a sum is called integration . In technical language, integral calculus studies a certain linear operator .
A quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. Numerical integration methods can generally be described as combining evaluations of the integrand to get an approximation to the integral.
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