Nilpotent ideals in polynomial and power series rings

Given a ring $R$ and polynomials $f(x),g(x)\in R[x]$ satisfying $f(x)Rg(x)=0$, we prove that the ideal generated by products of the coefficients of $f(x)$ and $g(x)$ is nilpotent. This result is generalized, and many well known facts, along with new ones, concerning nilpotent polynomials and power series are obtained. We also classify which of the standard nilpotence properties on ideals pass to polynomial rings or from ideals in polynomial rings to ideals of coefficients in base rings. In particular, we prove that if $I\leq R[x]$ is a left $T$-nilpotent ideal, then the ideal formed by the coefficients of polynomials in $I$ is also left $T$-nilpotent.