Elementary divisor rings and finitely presented modules

Larsen, Max D.; Lewis, William J.; Shores, Thomas S.

doi:10.1090/S0002-9947-1974-0335499-1

Elementary divisor rings and finitely presented modules
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by Max D. Larsen, William J. Lewis and Thomas S. Shores PDF

Trans. Amer. Math. Soc. 187 (1974), 231-248 Request permission

Abstract:

Throughout, rings are commutative with unit and modules are unital. We prove that R is an elementary divisor ring if and only if every finitely presented module over R is a direct sum of cyclic modules, thus providing a converse to a theorem of Kaplansky and answering a question of Warfield. We show that every Bezout ring with a finite number of minimal prime ideals is Hermite. So, in particular, semilocal Bezout rings are Hermite answering affirmatively a question of Henriksen. We show that every semihereditary Bezout ring is Hermite. Semilocal adequate rings are characterized and a partial converse to a theorem of Henriksen is established.

References

Eléments de mathématique

Algèbre commutative

Modules plats

36

Hans Fitting, Über den Zusammenhang zwischen dem Begriff der Gleichartigkeit zweier Ideale und dem Äquivalenzbegriff der Elementarteilertheorie, Math. Ann. 112 (1936), no. 1, 572–582 (German). MR 1513062, DOI 10.1007/BF01565430
Leonard Gillman and Melvin Henriksen, Rings of continuous functions in which every finitely generated ideal is principal, Trans. Amer. Math. Soc. 82 (1956), 366–391. MR 78980, DOI 10.1090/S0002-9947-1956-0078980-4
Leonard Gillman and Melvin Henriksen, Some remarks about elementary divisor rings, Trans. Amer. Math. Soc. 82 (1956), 362–365. MR 78979, DOI 10.1090/S0002-9947-1956-0078979-8
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
Olaf Helmer, The elementary divisor theorem for certain rings without chain condition, Bull. Amer. Math. Soc. 49 (1943), 225–236. MR 7744, DOI 10.1090/S0002-9904-1943-07886-X
Melvin Henriksen, Some remarks on elementary divisor rings. II, Michigan Math. J. 3 (1955/56), 159–163. MR 92772
Paul Jaffard, Les systèmes d’idéaux, Travaux et Recherches Mathématiques, IV, Dunod, Paris, 1960 (French). MR 0114810
C. U. Jensen, Arithmetical rings, Acta Math. Acad. Sci. Hungar. 17 (1966), 115–123. MR 190163, DOI 10.1007/BF02020446
Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
Irving Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949), 464–491. MR 31470, DOI 10.1090/S0002-9947-1949-0031470-3
Jean-Pierre Lafon, Anneaux locaux commutatifs sur lesquels tout module de type fini est somme directe de modules monogènes, J. Algebra 17 (1971), 575–591 (French, with English summary). MR 282968, DOI 10.1016/0021-8693(71)90010-X

Sur les anneaux commutatifs d’Hermite et à diviseurs élémentaires

273

44

Joachim Lambek, Lectures on rings and modules, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. With an appendix by Ian G. Connell. MR 0206032
Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
Marion Moore and Arthur Steger, Some results on completability in commutative rings, Pacific J. Math. 37 (1971), 453–460. MR 306182, DOI 10.2140/pjm.1971.37.453
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
Jack Ohm, Semi-valuations and groups of divisibility, Canadian J. Math. 21 (1969), 576–591. MR 242819, DOI 10.4153/CJM-1969-065-9
R. B. Warfield Jr., Decomposability of finitely presented modules, Proc. Amer. Math. Soc. 25 (1970), 167–172. MR 254030, DOI 10.1090/S0002-9939-1970-0254030-4
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