Commutative ideal theory without finiteness conditions: Primal ideals

Fuchs, Laszlo; Heinzer, William; Olberding, Bruce

doi:10.1090/S0002-9947-04-03583-4

Commutative ideal theory without finiteness conditions: Primal ideals
HTML articles powered by AMS MathViewer

by Laszlo Fuchs, William Heinzer and Bruce Olberding PDF

Trans. Amer. Math. Soc. 357 (2005), 2771-2798 Request permission

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: $A = \bigcap _{P \in \mathcal {X}_A}A_{(P)}$, 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.

References

Robert W. Berger, Various notions of associated prime ideals, Ann. Univ. Sarav. Ser. Math. 5 (1994), no. 3, i–iv and 245–271. MR 1293979
Nicolas Bourbaki, Commutative algebra. Chapters 1–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1972 edition. MR 979760
Marco Fontana, James A. Huckaba, and Ira J. Papick, Prüfer domains, Monographs and Textbooks in Pure and Applied Mathematics, vol. 203, Marcel Dekker, Inc., New York, 1997. MR 1413297
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
L. Fuchs, W. Heinzer and B. Olberding, Maximal prime divisors in arithmetical rings, to appear in Venezia 2002 Proceedings, Lecture Notes in Pure and Applied Math., Marcel Dekker.
Laszlo Fuchs and Edward Mosteig, Ideal theory in Prüfer domains—an unconventional approach, J. Algebra 252 (2002), no. 2, 411–430. MR 1925145, DOI 10.1016/S0021-8693(02)00040-6
R. Gilmer, A counterexample to two conjectures in ideal theory, Amer. Math. Monthly, 74 (1967), 195-197.
Robert W. Gilmer Jr. and William J. Heinzer, Overrings of Prüfer domains. II, J. Algebra 7 (1967), 281–302. MR 217055, DOI 10.1016/0021-8693(67)90073-7
Robert Gilmer and William Heinzer, Primary ideals with finitely generated radical in a commutative ring, Manuscripta Math. 78 (1993), no. 2, 201–221. MR 1202161, DOI 10.1007/BF02599309
Robert Gilmer and William Heinzer, Imbeddability of a commutative ring in a finite-dimensional ring, Manuscripta Math. 84 (1994), no. 3-4, 401–414. MR 1291129, DOI 10.1007/BF02567465
William Heinzer, Quotient overrings of integral domains, Mathematika 17 (1970), 139–148. MR 265334, DOI 10.1112/S0025579300002813
William Heinzer and David Lantz, The Laskerian property in commutative rings, J. Algebra 72 (1981), no. 1, 101–114. MR 634618, DOI 10.1016/0021-8693(81)90313-6
William Heinzer and Jack Ohm, Locally noetherian commutative rings, Trans. Amer. Math. Soc. 158 (1971), 273–284. MR 280472, DOI 10.1090/S0002-9947-1971-0280472-2
William Heinzer and Sylvia Wiegand, Prime ideals in two-dimensional polynomial rings, Proc. Amer. Math. Soc. 107 (1989), no. 3, 577–586. MR 982402, DOI 10.1090/S0002-9939-1989-0982402-3
Juana Iroz and David E. Rush, Associated prime ideals in non-Noetherian rings, Canad. J. Math. 36 (1984), no. 2, 344–360. MR 749988, DOI 10.4153/CJM-1984-021-6
C. U. Jensen, Arithmetical rings, Acta Math. Acad. Sci. Hungar. 17 (1966), 115–123. MR 190163, DOI 10.1007/BF02020446
Irving Kaplansky, Commutative rings, Revised edition, University of Chicago Press, Chicago, Ill.-London, 1974. MR 0345945
W. Krull, Idealtheorie in Ringen ohne Endlichkeitsbedingung, Math. Ann. 101 (1929), 729-744.
Wolfgang Krull, Idealtheorie, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 46, Springer-Verlag, Berlin-New York, 1968 (German). Zweite, ergänzte Auflage. MR 0229623, DOI 10.1007/978-3-642-87033-0
Richard A. Kuntz, Associated prime divisors in the sense of Krull, Canadian J. Math. 24 (1972), 808–818. MR 313233, DOI 10.4153/CJM-1972-079-0
Donald L. McQuillan, Rings of integer-valued polynomials determined by finite sets, Proc. Roy. Irish Acad. Sect. A 85 (1985), no. 2, 177–184. MR 845540
S. Mori, Über eindeutige Reduktion in Ringen ohne Teilerkettensatz, J. Sci. Hiroshima Univ. A 3 (1933), 275-318.
Masayoshi Nagata, Some remarks on prime divisors, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 33 (1960/61), 297–299. MR 118747, DOI 10.1215/kjm/1250775914
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
E. Noether, Idealtheorie in Ringbereichen, Math. Ann. 83 (1921), 24-66.
Jack Ohm and R. L. Pendleton, Rings with noetherian spectrum, Duke Math. J. 35 (1968), 631–639. MR 229627
Bruce Olberding, Globalizing local properties of Prüfer domains, J. Algebra 205 (1998), no. 2, 480–504. MR 1632741, DOI 10.1006/jabr.1997.7406
D. Rush and L. Wallace, Noetherian maximal spectrum and coprimely packed localizations of polynomial rings, Houston J. Math. 28 (2002) 437-448.
Douglas H. Underwood, On some uniqueness questions in primary representations of ideals, J. Math. Kyoto Univ. 9 (1969), 69–94. MR 246865, DOI 10.1215/kjm/1250524012
Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0120249, DOI 10.1007/978-3-662-29244-0