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 for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by . 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 . 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. Primary decomposition of an ideal Let be a Noetherian commutative ring. An ideal of is called primary if it is a proper ideal and for each pair of elements and in such that is in , either or some power of is in ; equivalently, every zero-divisor in the quotient is nilpotent. The radical of a primary ideal is a prime ideal and is said to be -primary for . Let be an ideal in . Then has an irredundant primary decomposition into primary ideals: . Irredundancy means: Removing any of the changes the intersection, i.e. for each we have: . The prime ideals are all distinct. Moreover, this decomposition is unique in the two ways: The set is uniquely determined by ,