Jordan Blocks, Differential Traces, and Colon-Ideal Noncontainment over Noetherian Quotients

Let $k$ be a field of characteristic zero, let $R$ be a commutative $k$-algebra, let $I$ be a proper ideal of $R$, and assume that $A=R/I$ is Noetherian. We prove that if $a=[f]$ in $A$ is nilpotent and $d_{A/k}a=0$, then $I:f$ is not contained in $(I,f)$. If $R$ is local with maximal ideal $\mathfr