Editing Primary decomposition (section)

{{short description|In algebra, expression of an ideal as the intersection of ideals of a specific type}}
In [[mathematics]], the '''Lasker–Noether theorem''' states that every [[Noetherian ring]] is a '''Lasker ring''', which means that every ideal can be decomposed as an intersection, called '''primary decomposition''', of finitely many ''[[primary ideal]]s'' (which are related to, but not quite the same as, powers of [[prime ideal]]s).  The theorem was first proven by {{harvs|txt|authorlink=Emanuel Lasker|first=Emanuel|last= Lasker|year= 1905}} for the special case of [[polynomial ring]]s and convergent [[power series]] rings, and was proven in its full generality by {{harvs|txt|authorlink=Emmy Noether|first=Emmy |last=Noether|year= 1921}}.

The Lasker–Noether theorem is an extension of the [[fundamental theorem of arithmetic]], and more generally the [[fundamental theorem of finitely generated abelian groups]] to all Noetherian rings. The theorem plays an important role in [[algebraic geometry]], by asserting that every [[algebraic set]] may be uniquely decomposed into a finite union of [[irreducible component]]s.

It has a straightforward extension to [[Module (mathematics)|modules]] stating that every submodule of a [[finitely generated module]] over a Noetherian ring is a finite intersection of primary submodules. This contains the case for rings as a special case, considering the ring as a module over itself, so that ideals are submodules. This also generalizes the primary decomposition form of the [[structure theorem for finitely generated modules over a principal ideal domain]], and for the special case of polynomial rings over a field, it generalizes the decomposition of an algebraic set into a finite union of (irreducible) varieties.

The first algorithm for computing primary decompositions for polynomial rings over a field of [[Characteristic (algebra)|characteristic]] 0<ref group="Note">Primary decomposition requires testing irreducibility of polynomials, which is not always algorithmically possible in nonzero characteristic.</ref> was published by Noether's student {{harvs|txt|authorlink=Grete Hermann|first=Grete |last=Hermann|year= 1926|pg=89}}.<ref>{{cite book|editor1-last=Ciliberto|editor1-first=Ciro|editor2-last=Hirzebruch|editor2-first=Friedrich|editor3-last=Miranda|editor3-first=Rick|editor4-last=Teicher|editor4-first=Mina|editor4-link= Mina Teicher |title=Applications of Algebraic Geometry to Coding Theory, Physics and Computation|date=2001|publisher=Springer Netherlands|location=Dordrecht|isbn=978-94-010-1011-5|url=https://www.springer.com/us/book/9781402000041|language=en}}</ref><ref>{{cite journal|title=Die Frage der endlich vielen Schritte in der Theorie der Polynomideale|author=Hermann, G.|journal=Mathematische Annalen|volume=95|year=1926|pages=736–788|url=https://eudml.org/doc/159153|language=de|doi=10.1007/BF01206635|s2cid=115897210 }}</ref> The decomposition does not hold in general for non-commutative Noetherian rings. Noether gave an example of a non-commutative Noetherian ring with a right ideal that is not an intersection of primary ideals.