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Proceedings of the American Mathematical Society

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

 

 

Zero divisors and nilpotent elements in power series rings


Author: David E. Fields
Journal: Proc. Amer. Math. Soc. 27 (1971), 427-433
MSC: Primary 13.93
DOI: https://doi.org/10.1090/S0002-9939-1971-0271100-6
MathSciNet review: 0271100
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Abstract: It is well known that a polynomial $ f(X)$ over a commutative ring $ R$ with identity is nilpotent if and only if each coefficient of $ f(X)$ is nilpotent; and that $ f(X)$ is a zero divisor in $ R[X]$ if and only if $ f(X)$ is annihilated by a nonzero element of $ R$. This paper considers the problem of determining when a power series $ g(X)$ over $ R$ is either nilpotent or a zero divisor in $ R[[X]]$. If $ R$ is Noetherian, then $ g(X)$ is nilpotent if and only if each coefficient of $ g(X)$ is nilpotent; and $ g(X)$ is a zero divisor in $ R[[X]]$ if and only if $ g(X)$ is annihilated by a nonzero element of $ R$. If $ R$ has positive characteristic, then $ g(X)$ is nilpotent if and only if each coefficient of $ g(X)$ is nilpotent and there is an upper bound on the orders of nilpotency of the coefficients of $ g(X)$. Examples illustrate, however, that in general $ g(X)$ need not be nilpotent if there is an upper bound on the orders of nilpotency of the coefficients of $ g(X)$, and that $ g(X)$ may be a zero divisor in $ R[[X]]$ while $ g(X)$ has a unit coefficient.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0271100-6
Keywords: Formal power series, zero divisors, nilpotent elements
Article copyright: © Copyright 1971 American Mathematical Society