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Proceedings of the American Mathematical Society

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

 

 

On the primitivity of polynomial rings with nonprimitive coefficient rings


Author: T. J. Hodges
Journal: Proc. Amer. Math. Soc. 98 (1986), 553-558
MSC: Primary 16A20
MathSciNet review: 861748
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Abstract: For a hereditary noetherian prime ring $ R$ with classical quotient ring $ Q$, various necessary and sufficient conditions are given for the polynomial ring $ R[{X_1}, \ldots ,{X_n}]$ to be primitive when $ R$ itself is not primitive. It is shown that if $ R$ is a local hereditary noetherian prime ring, then $ R[X]$ is primitive if and only if $ Q[X]$ is primitive. Similarly, for a semilocal hereditary noetherian prime ring $ R$ whose Jacobson radical contains a nonzero central element, it is shown that $ R[{X_1}, \ldots ,{X_n}]$ is primitive if and only if $ Q[{X_1}, \ldots ,{X_n}]$ is primitive.


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DOI: https://doi.org/10.1090/S0002-9939-1986-0861748-0
Keywords: Primitivity, polynomial rings, hereditary noetherian prime rings
Article copyright: © Copyright 1986 American Mathematical Society