Commutative algebra | Algebraic geometry | Theorems in ring theory

Primary decomposition

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 ideals (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by Emanuel Lasker for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by Emmy Noether. 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 components. It has a straightforward extension to 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 0 was published by Noether's student Grete Hermann. 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. (Wikipedia).

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Michael Atiyah | Tertiary ideal | Prime ideal | Quotient ring | Irreducible component | Support of a module | Zero of a function | Minimal prime ideal | Annihilator (ring theory) | Associated prime | Monomial | Homogeneous polynomial | Radical of an ideal | Resultant | Polynomial ring | Algebraic set | Flat module | Intersection theory | Mathematics | Unit (ring theory) | Field (mathematics) | Fundamental theorem of arithmetic | Algebraic geometry | Noetherian ring | Primary ideal | Complete intersection | Structure theorem for finitely generated modules over a principal ideal domain | Generic property | Nicolas Bourbaki | Polynomial greatest common divisor | Unique factorization domain | Fundamental theorem of finitely generated abelian groups | Finitely generated module | Principal ideal | Module (mathematics)