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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Elementary divisor rings and finitely presented modules


Authors: Max D. Larsen, William J. Lewis and Thomas S. Shores
Journal: Trans. Amer. Math. Soc. 187 (1974), 231-248
MSC: Primary 13F05
DOI: https://doi.org/10.1090/S0002-9947-1974-0335499-1
MathSciNet review: 0335499
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0335499-1
Keywords: Bezout rings, finitely presented modules, Hermite rings, elementary divisor rings, adequate rings
Article copyright: © Copyright 1974 American Mathematical Society