Commutative ideal theory without finiteness conditions: Primal ideals

Our goal is to establish an efficient decomposition of an ideal $A$ of a commutative ring $R$ as an intersection of primal ideals. We prove the existence of a canonical primal decomposition: $A = \bigcap_{P \in \mathcal{X}_A}A_{(P)}$, where the $A_{(P)}$ are isolated components of $A$ that are primal ideals having distinct and incomparable adjoint primes $P$. For this purpose we define the set $\operatorname{Ass}(A)$ of associated primes of the ideal $A$ 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 $A$ as an irredundant intersection of isolated components of $A$. Using our canonical primal decomposition, we obtain an affirmative answer to a question raised by Fuchs, and also prove for $P \in \operatorname{Spec}R$ that an ideal $A \subseteq P$ is an intersection of $P$-primal ideals if and only if the elements of $R \setminus P$ are prime to $A$. We prove that the following conditions are equivalent: (i) the ring $R$ is arithmetical, (ii) every primal ideal of $R$ is irreducible, (iii) each proper ideal of $R$ is an intersection of its irreducible isolated components. We classify the rings for which the canonical primal decomposition of each proper ideal is an irredundant decomposition of irreducible ideals as precisely the arithmetical rings with Noetherian maximal spectrum. In particular, the integral domains having these equivalent properties are the Prüfer domains possessing a certain property.