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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Right-bounded factors in an LCM domain


Author: Raymond A. Beauregard
Journal: Trans. Amer. Math. Soc. 200 (1974), 251-266
MSC: Primary 16A02
DOI: https://doi.org/10.1090/S0002-9947-1974-0379553-7
MathSciNet review: 0379553
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Abstract: A right-bounded factor is an element in a ring that generates a right ideal which contains a nonzero two-sided ideal. Right-bounded factors in an LCM domain are considered as a generalization of the theory of two-sided bounded factors in an atomic $ 2$-fir, that is, a weak Bezout domain satisfying the acc and dcc for left factors. Although some elementary properties are valid in a more general context most of the main results are obtained for an LCM domain satisfying $ ({\text{M}})$ and the dcc for left factors; the condition $ ({\text{M}})$ is imposed to insure that prime factorizations are unique in an appropriate sense. The right bound $ {b^ \ast }$ of a right bounded element $ b$ is considered in general, then in case $ b$ is a prime, and finally in case $ b$ is indecomposable. The effect of assuming that right bounds are two-sided is also considered.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0379553-7
Keywords: Bounded right ideal, right-invariant factor, prime factorization, indecomposable element, LCM domain, $ 2$-fir, principal ideal domain
Article copyright: © Copyright 1974 American Mathematical Society