Let R be a principal ideal domain.

(a) Every proper ideal is a product P₁P₂· • •P_n of maximal ideals, which are uniquely determined up to order.

(b) An ideal P in R is said to be primary if ab ϵ P and a ϵ P imply bⁿ ϵ P for some n. Show that P is primary if and only if for some n, P = (pⁿ), where p ϵ R is prime ( = irreducible) or p = 0.

(c) If P₁ ,P₂ , ••• , P_n are primary ideals such that P_i=(p_iⁿⁱ)and the p_i are distinct primes, then P₁P₂· · ·P_n = P₁ ∩ P₂ ∩ · · · ∩ P_n.

(d) Every proper ideal in R can be expressed (uniquely up to order) as the intersection of a finite number of primary ideals.