# Primary decomposition

A representation of an ideal $I$ of a ring $R$ (or of a submodule $N$ of a module $M$) as an intersection of primary ideals (primary submodules, cf. Primary ideal). The primary decomposition generalizes the factorization of an integer into a product of powers of distinct prime numbers. The existence of primary decompositions in a polynomial ring was proved by E. Lasker [1], and in an arbitrary commutative Noetherian ring by E. Noether [2]. Let $R$ be a commutative Noetherian ring. A primary decomposition $I = \cap_{i=1}^n Q_i$ is called irreducible if $\cap_{i\ne j}Q_i \ne I$ for any $j = 1,\ldots,n$ and if the radicals $P_1,\ldots,P_n$ of the ideals $Q_1,\ldots,Q_n$ are pairwise distinct (the radical of a primary ideal $Q$ is the unique prime ideal $P$ such that $P^n \subseteq Q$ for some natural number $n$). The set of prime ideals $\{P_1,\ldots,P_n\}$ is uniquely determined by the ideal $I$ (the first uniqueness theorem for primary decompositions). The minimal elements (with respect to inclusion) of this set are called the isolated prime ideals of $I$, the other elements are called the imbedded prime ideals. The primary ideals corresponding to isolated prime ideals are also uniquely determined by $I$ (the second uniqueness theorem for primary decompositions, cf. [3]). The isolated prime ideals of an ideal $I$ of a polynomial ring over a field correspond to the irreducible components of the affine variety of roots of $I$. There are various generalizations of the notion of primary decomposition. The axiomatization of primary decompositions led to the development of the additive theory of ideals.