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 over a commutative ring with identity is nilpotent if and only if each coefficient of is nilpotent; and that is a zero divisor in if and only if is annihilated by a nonzero element of . This paper considers the problem of determining when a power series over is either nilpotent or a zero divisor in . If is Noetherian, then is nilpotent if and only if each coefficient of is nilpotent; and is a zero divisor in if and only if is annihilated by a nonzero element of . If has positive characteristic, then is nilpotent if and only if each coefficient of is nilpotent and there is an upper bound on the orders of nilpotency of the coefficients of . Examples illustrate, however, that in general need not be nilpotent if there is an upper bound on the orders of nilpotency of the coefficients of , and that may be a zero divisor in while has a unit coefficient.

**[1]**N. H. McCoy,*Remarks on divisors of zero*, Amer. Math. Monthly**49**(1942), 286–295. MR**0006150**, https://doi.org/10.2307/2303094**[2]**Neal H. McCoy,*The theory of rings*, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1964. MR**0188241****[3]**Oscar Zariski and Pierre Samuel,*Commutative algebra, Volume I*, The University Series in Higher Mathematics, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR**0090581**

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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