Left and right invariance in an integral domain

Author:
Raymond A. Beauregard

Journal:
Proc. Amer. Math. Soc. **67** (1977), 201-205

MSC:
Primary 16A02

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.

**[1]**Raymond A. Beauregard,*Right 𝐿𝐶𝑀 domains*, Proc. Amer. Math. Soc.**30**(1971), 1–7. MR**0279125**, 10.1090/S0002-9939-1971-0279125-1**[2]**Raymond A. Beauregard,*Right-bounded factors in an LCM domain*, Trans. Amer. Math. Soc.**200**(1974), 251–266. MR**0379553**, 10.1090/S0002-9947-1974-0379553-7**[3]**P. M. Cohn,*Free rings and their relations*, Academic Press, London-New York, 1971. London Mathematical Society Monographs, No. 2. MR**0371938****[4]**Nathan Jacobson,*The Theory of Rings*, American Mathematical Society Mathematical Surveys, vol. II, American Mathematical Society, New York, 1943. MR**0008601****[5]**Arun Vinayak Jategaonkar,*Left principal ideal domains*, J. Algebra**8**(1968), 148–155. MR**0218387**

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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