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Some Criteria for Hermite Rings and Elementary Divisor Rings

Published online by Cambridge University Press:  20 November 2018

Thomas S. Shores  and
Roger Wiegand
Thomas S. Shores
Affiliation:
University of Nebraska-Lincoln, Lincoln, Nebraska
Roger Wiegand
Affiliation:
University of Nebraska-Lincoln, Lincoln, Nebraska
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Recall that a ring R (all rings considered are commutative with unit) is an elementary divisor ring (respectively, a Hermite ring) provided every matrix over R is equivalent to a diagonal matrix (respectively, a triangular matrix). Thus, every elementary divisor ring is Hermite, and it is easily seen that a Hermite ring is Bezout, that is, finitely generated ideals are principal. Examples have been given [4] to show that neither implication is reversible.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 6 , December 1974 , pp. 1380 - 1383
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Endo, S., On semi-hereditary rings, J. Math. Soc. Japan 13 (1961), 109–119.Google Scholar
2. Estes, D. and Ohm, J., Stable range in commutative rings, J. Algebra 7 (1967), 343–362.Google Scholar
3. Gillman, L. and Henriksen, M., Some remarks about elementary divisor rings, Trans. Amer. Math. Soc. 82 (1956), 362–365.Google Scholar
4. Gillman, L. and Henriksen, M., Rings of continuous functions in which every finitely generated ideal is principal, Trans. Amer. Math. Soc. 82 (1956), 366–391.Google Scholar
5. Henriksen, M., Some remarks on elementary divisor rings, II, Michigan Math. J. 8 (1955-56), 159–163.Google Scholar
6. Heinzer, W., J-noetherian integral domains with 1 in the stable range, Proc. Amer. Math. Soc. 19 (1968), 1369–1372.Google Scholar
7. Kaplansky, I., Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949), 464–491.Google Scholar
8. Larsen, M., Lewis, W., and Shores, T., Elementary divisor rings and finitely presented modules, Trans. Amer. Math. Soc. 187(1974), 231–248.Google Scholar
9. Ohm, J. and Pendleton, R. L., Rings with noetherian spectrum, Duke Math. J. 85 (1968), 631–640.Google Scholar
10. Ohm, J. and Pendleton, R. L., Addendum to ‘Rings with noetherian spectrum', Duke Math. J. 35 (1968), 875.Google Scholar
11. Quentel, Y., Sur la compacité du spectre minimal d'un anneau, Bull. Soc. Math. France 99 (1971), 265–272.Google Scholar
12. Shores, T., The structure of Loewy modules, J. Reine Angew. Math. 254 (1972), 204–220.Google Scholar