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Proceedings of the American Mathematical Society

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

 

 

Order in a special class of rings and a structure theorem


Author: Alexander Abian
Journal: Proc. Amer. Math. Soc. 52 (1975), 45-49
MSC: Primary 17E05; Secondary 06A70
DOI: https://doi.org/10.1090/S0002-9939-1975-0374222-8
Addendum: Proc. Amer. Math. Soc. 61 (1976), 188.
MathSciNet review: 0374222
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Abstract: Below a special class of not necessarily associative or commutative rings $ A$ is considered which is characterized by the property that $ A$ has no nonzero nilpotent element and that a product of elements of $ A$ which is equal to zero remains equal to zero no matter how its factors are associated. It is shown that $ (A, \leqslant )$ is a partially ordered set where $ x \leqslant y$ if and only if $ xy = {x^2}$. Also it is shown that $ (A, \leqslant )$ is infinitely distributive, i.e., $ r\sup {x_i} = \sup r{x_i}$. Finally, based on Zorn's lemma it is shown that $ A$ is isomorphic to a subdirect product of not necessarily associative or commutative rings without zero divisors.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0374222-8
Keywords: Nilpotent, partial order, infinite distributivity
Article copyright: © Copyright 1975 American Mathematical Society