Left Ore principal right ideal domains

Author:
Raymond A. Beauregard

Journal:
Proc. Amer. Math. Soc. **102** (1988), 459-462

MSC:
Primary 16A02

DOI:
https://doi.org/10.1090/S0002-9939-1988-0928960-5

MathSciNet review:
928960

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Abstract | References | Similar Articles | Additional Information

Abstract: The rings referred to in the title of this paper have long been conjectured to be principal left ideal domains. In a recent paper [**6**] Cohn and Schofield have produced examples (of simple rings) showing that this is not the case. As a result, interest in this type of ring has deepened (see [**5**], for example, where these rings are referred to as right principal Bezout domains). Our main purpose here is to prove that such rings are either principal left ideal domains or left and right primitive rings.

**[1]**R. A. Beauregard,*Infinite primes and unique factorization in a principal right ideal domain*, Trans. Amer. Math. Soc.**141**(1969), 245-253. MR**0242879 (39:4206)****[2]**-,*Right bounded factors in an LCM domain*, Trans. Amer. Math. Soc.**200**(1974), 251-266. MR**0379553 (52:458)****[3]**-,*Left and right invariance in an integral domain*, Proc. Amer. Math. Soc.**67**(1977), 201-205. MR**0457480 (56:15685)****[4]**P. M. Cohn,*Free rings and their relations*, 2nd ed., Academic Press, London, New York, 1985. MR**800091 (87e:16006)****[5]**-,*Right principal Bezout domains*, J. London Math. Soc. (2)**35**(1987), 251-262. MR**881514 (88f:16001)****[6]**P. M. Cohn and A. H. Schofield,*Two examples of principal ideal domains*, Bull. London Math. Soc.**17**(1985), 25-28. MR**766442 (86b:16001)****[7]**N. Jacobson,*Structure of rings*(revised), Amer. Math. Soc. Colloq Publ., vol. 37, Providence, R.I., 1964. MR**0222106 (36:5158)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0928960-5

Keywords:
Principal ideal domain,
Ore domain,
Bezout domain,
invariant element,
bounded element,
primitive ring

Article copyright:
© Copyright 1988
American Mathematical Society