Abstract
It is proved that if R is a commutative ring in which zero is an adequate element, then R is an exchange ring and every zero adequate ring is an exchange ring. There is a new description of adequate rings; this is an answer to questions formulated by Larsen, Lewis, and Shores.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. Bourbaki, Commutative Algebra [Russian translation], Mir, Moscow (1971).
V. Camillo and H. P. Yu, “Exchange rings, units and idempotents,” Commun. Algebra, 22, No. 12, 4737–4749 (1994).
O. Helmer, “The elementary divisor for certain rings without chain condition,” Bull. Am. Math. Soc., 49, No. 2, 225–236 (1943).
M. D. Larsen,W. J. Lewis, and T. S. Shores, “Elementary divisor rings and finitely presented modules,” Trans. Am. Math. Soc., 187, 231–248 (1974).
W. K. Nicholson, “Lifting idempotents and exchange rings,” Trans. Am. Math. Soc., 229, 269–278 (1977).
B. V. Zabavsky, “Elementary divisors of adequate rings with finite minimal prime ideals,” in: Algebra and Topology [in Ukrainian], LGU, Lviv (1996), pp. 74–78.
B. Zabavsky and M. Komarnitsky, “On adequate rings,” Visn. Lviv Univ., 39–43 (1988).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 3, pp. 61–66, 2011/12.
Rights and permissions
About this article
Cite this article
Zabavsky, B.V., Bilavska, S.I. Every zero adequate ring is an exchange ring. J Math Sci 187, 153–156 (2012). https://doi.org/10.1007/s10958-012-1058-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-012-1058-y