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Lasker–Noether theorem
From Wikipedia, the free encyclopedia
In mathematics, the Lasker–Noether theorem provides a vast generalization of the fundamental theorem of arithmetic to embrace the rings of algebraic geometry. The theorem was first proven by Emanuel Lasker in 1905 for the special case of polynomial rings. It was proven in its full generality by Emmy Noether in 1921.
The Lasker–Noether theorem states that every ideal of a Noetherian ring has a primary decomposition: it can be written as an intersection of finitely many primary ideals. Usually, the primary decomposition is written as an intersection of primary ideals, none of which contain the intersection of the other ones, which all have distinct associated prime ideals.
Lasker's original 1905 proof, which appeared as Zur Theorie der Moduln und Ideale in Mathematische Annalen, established the existence of primary decompositions for polynomial rings. This is the most important special case, since it includes the coordinate rings of affine varieties.
In her 1921 paper, Idealtheorie in Ringbereichen, Noether introduced the ascending chain condition for ideals, and demonstrated that the existence of a primary decomposition follows for any commutative ring that satisfies the ascending chain condition. (Rings satisfying the ascending chain condition on ideals are now known as Noetherian rings.)
The first algorithm for computing primary decompositions for polynomial rings was published by Noether's student Grete Hermann in 1926.
- Eisenbud, David (1995), Commutative algebra, vol. 150, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, MR1322960, ISBN 978-0-387-94268-1; 978-0-387-94269-8, esp. section 3.3.