In
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, a commutative
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
''A'' is said to have the approximation property with respect to an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
''I'' if each finite system of polynomial equations with coefficients in ''A'' has a solution in ''A'' if and only if it has a solution in the
''I''-adic completion of ''A''.
[ The notion of the approximation property is due to ]Michael Artin
Michael Artin (; born 28 June 1934) is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology Mathematics Department, known for his contributions to algebraic geometry. .
See also
* Artin approximation theorem
*Popescu's theorem In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu,
states:
:Let ''A'' be a Noetherian ring and ''B'' a Noetherian algebra over it. Then, the structure map ''A'' → ''B'' is a regular homomorphism if and ...
Notes
References
*
*
*
*
Ring theory
{{algebra-stub