Zero divisors and nilpotent elements in power series rings

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.