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Proceedings of the American Mathematical Society

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

 

 

Rings satisfying monomial constraints


Authors: Mohan S. Putcha and Adil Yaqub
Journal: Proc. Amer. Math. Soc. 39 (1973), 10-18
MSC: Primary 16A38
MathSciNet review: 0313306
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Abstract: The following theorem is proved: Suppose $ R$ is an associative ring and suppose $ J$ is the Jacobson radical of $ R$. Suppose that for all $ {x_1}, \cdots ,{x_n}$ in $ R$, there exists a word $ {w_{{x_1}}}, \cdots ,{x_n}({x_1}, \cdots ,{x_n})$, depending on $ {x_1}, \cdots ,{x_n}$, in which at least one $ {x_i}$ ($ i$ varies) is missing, and such that $ {x_1} \cdots {x_n} = {w_{{x_1}, \cdots ,{x_n}}}({x_1}, \cdots ,{x_n})$. Then $ J$ is a nil ring of bounded index and $ R/J$ is finite. It is further proved that a commutative nil semigroup satisfies the above identity if and only if it is nilpotent.


References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/S0002-9939-1973-0313306-5
Article copyright: © Copyright 1973 American Mathematical Society