Zero divisors and nilpotent elements in power series rings
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- by David E. Fields
- Proc. Amer. Math. Soc. 27 (1971), 427-433
- DOI: https://doi.org/10.1090/S0002-9939-1971-0271100-6
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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.References
- N. H. McCoy, Remarks on divisors of zero, Amer. Math. Monthly 49 (1942), 286–295. MR 6150, DOI 10.2307/2303094
- Neal H. McCoy, The theory of rings, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1964. MR 0188241
- Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- 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