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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Left and right invariance in an integral domain


Author: Raymond A. Beauregard
Journal: Proc. Amer. Math. Soc. 67 (1977), 201-205
MSC: Primary 16A02
DOI: https://doi.org/10.1090/S0002-9939-1977-0457480-2
MathSciNet review: 0457480
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Abstract: A ring is said to be right (left) invariant if each of its right (left) ideals is twosided. In this paper we resolve the conjecture: Every right invariant integral domain which satisfies the left Ore (multiple) condition is left invariant. A proof is given for the class of LCM domains satisfying a finiteness condition. An example is given to show that the LCM hypothesis cannot be dropped. A second example shows that the conjecture fails even in a Bezout domain which does not have the finiteness condition. The problem of right versus left boundedness is also considered.


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DOI: https://doi.org/10.1090/S0002-9939-1977-0457480-2
Keywords: Invariance, LCM domain, Bezout domain, Ore domain, boundedness, PRI domain, left-right symmetry
Article copyright: © Copyright 1977 American Mathematical Society