A ring is Noetherian if any ascending chain of ideals I₁ ⊆ I₂ ⊆ ⋯ ⊆ I_k ⊆ ⋯ eventually becomes constant. The condition guarantees finite decomposition in different senses, for example decomposing varieties into finite irreducible components or, in ℤ, to factorization of integers into primes. Principal ideal domains are Noetherian. See Hilbert’s basis theorem.