Artinian decomposition into local factors
= Artinian decomposition into local factors
{c}
{title2=$R\cong\prod_iR_i$}
Every nonzero commutative <Artinian ring> is a finite product of <Artinian local rings>. The primary components of zero have distinct maximal radicals and are pairwise <comaximal ideals>, so the <Chinese remainder theorem> gives the product. Each local factor has nilpotent maximal ideal and finite <composition length>. These factors and the components of zero are unique.