Let $R$ be a field and $I$ be the set of non units in $R[[x]]$. Prove $I$ is an ideal.
So far, I have that an element of $R[[x]]$ is a unit iff its constant term is a unit in $R$ because for any constant term which is a unit its inverse must exist. But if $R$ is field wouldn't all the constant terms be units? Thus $I = \{0\}$, which seemed to be incorrect.
In a field there is a non-unit, namely $0$. It follows that $I$ is the set of power series with constant term $0$. You should be able to show that this is an ideal in $R[[x]]$.