TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 7, Pages 2771–2798 S 0002-9947(04)03583-4 Article electronically published on September 2, 2004 COMMUTATIVE IDEAL THEORY WITHOUT FINITENESS CONDITIONS: PRIMAL IDEALS LASZLO FUCHS, WILLIAM HEINZER, AND BRUCE OLBERDING Abstract. 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 =  P ∈X A A (P ) , where the A (P ) are isolated components of A that are primal ideals having distinct and incompa- rable adjoint primes P . For this purpose we define the set 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 residu- ally 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 ∈ Spec R that an ideal A ⊆ P is an intersection of P -primal ideals if and only if the elements of R \ P are prime to A. We prove that the following con- ditions 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 de- composition 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 Pr¨ ufer domains possessing a certain property. Introduction In her seminal paper [27], Emmy Noether proved that in a commutative ring satisfying the ascending chain condition on ideals every ideal is the intersection of a finite number of irreducible ideals, and the irreducible ideals are primary ideals. She went on to establish several intersection decompositions, one of which featured primary ideal components with distinct radicals. Among rings without the ascending chain condition, the rings in which such a decomposition holds for all ideals (called Laskerian rings) are few and far between (see, e.g., Heinzer and Lantz [13]). One reason for this is that irreducible ideals need not be primary. Thus, if we wish to obtain analogous decompositions in general rings, then we have to replace primary ideals by more general ideals. Natural candidates are the primal ideals introduced in Fuchs [5], where it is shown that every ideal is the intersection of (in general infinitely many) irreducible ideals, Received by the editors January 2, 2003 and, in revised form, November 4, 2003. 2000 Mathematics Subject Classification. Primary 13A15, 13F05. Key words and phrases. Primal ideal, associated prime, arithmetical ring, Pr¨ ufer domain. c 2004 American Mathematical Society 2771 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use