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

each ie ω0} and we define A R [[X]] to be the ideal of R [[X]] which is generated by A. Then A R [[X]] = {f(X): Af S B for some finitely generated ideal B of R with B £ A}. It is clear that A iϋ [[X]] S A [[X]]; equality holds if and only if each countably generated ideal of R contained in A is contained in a finitely generated ideal of R contained in A. In particular, if V is a valuation ring containing an ideal A which is countably generated but not finitely generated, then A V[[X]] c A [[X]]. Finally, we note that if A is an ideal of i2, then R [[X]]/A [[X]] ~ (R/A) [[X]]; hence A [[X]] is a prime ideal of R [[X]] if and only if A is a prime ideal of R.

(1958). Teaching the Accountant's Role in Budgeting. The Journal of Business Education: Vol. 33, No. 8, pp. 325-326.

SUMMARY A serologically distinct infectious bronchitis virus was isolated from clinically affected broilers in Arkansas. In each case the birds had been vaccinated 30 to 50 days previously with commercial Newcastle-bronchitis vaccines. The clinical history and isolation methods are described.