Primary decomposition theorem for ideals

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Statement

Let R be a Noetherian ring, and I be a proper ideal in R. Then I admits a primary decomposition, viz., there exists a finite collection Q_1, Q_2, \ldots, Q_n of primary ideals such that:

I = \bigcap_{i=1}^n Q_i

Further, the set of associated primes for I (viewed as a R-module) is the same as the set of radicals for the Q_is.

Also see primary decomposition theorem for modules.