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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.
Formal statement
The Lasker-Noether theorem states that every ideal of a Noetherian ring has a primary decomposition: it can be written as the intersection of primary ideals.
History
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.
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