Commutative ideal theory without finiteness conditions: Primal ideals

Our goal is to establish an efficient decomposition of an ideal A A of a commutative ring R R as an intersection of primal ideals. We prove the existence of a canonical primal decomposition: A = ⋂ P ∈ X A A ( P ) A = P ∈ X_AA(P) , where the A ( P ) A(P) are isolated components of A A that are primal ideals having distinct and incomparable adjoint primes P P . For this purpose we define the set Ass ⁡ ( A ) Ass(A) of associated primes of the ideal A A to be those defined and studied by Krull. We determine conditions for the canonical primal decomposition to be irredundant, or residually maximal, or the unique representation of A A as an irredundant intersection of isolated components of A A . Using our canonical primal decomposition, we obtain an affirmative answer to a question raised by Fuchs, and also prove for P ∈ Spec ⁡ R P ∈ SpecR that an ideal A ⊆ P A ⊆ P is an intersection of <