  Let $R$ be a commutative ring, $\mathcal A$ an $R$-algebra (not necessarily
commutative) and $V$ an $R$-subspace or $R$-submodule of $\mathcal A$. By the
radical of $V$ we mean the set of all elements $a\in \mathcal A$ such that
$a^m\in V$ for all $m\gg 0$. We derive (and show) some necessary conditions
satisfied by the elements in the radicals of the kernel of some (partial)
differential operators, such as all differential operators of commutative
algebras; the differential operators $P(D)$ of (noncommutative) $\mathcal A$
with certain conditions, where $P(\cdot)$ is a polynomial in $n$ commutative
free variables and $D=(D_1, D_2, \dots, D_n)$ are either commuting locally
finite $R$-derivations or commuting $R$-derivations of $\mathcal A$ such that
for each $1\le i\le n$, $\mathcal A$ can be decomposed as a direct sum of the
generalized eigen-subspaces of $D_i$; etc. In particular, we show that the
kernel of certain differential operators of $\mathcal A$ is a Mathieu subspace
(see \cite{GIC, MS}) of $\mathcal A$. We then apply some results above to study
$R$-derivations of $\mathcal A$, which are locally algebraic or locally
integral over $R$. In particular, we show that if $R$ is an integral domain of
characteristic zero and $\mathcal A$ is reduced and torsion-free as an
$R$-module, then $\mathcal A$ has no nonzero locally algebraic $R$-derivations.
We also show a formula for the determinant of a differential vandemonde matrix
over a commutative algebra $\mathcal A$. This formula not only provides some
information for the elements in the radical of the kernel of all ordinary
differential operators of $\mathcal A$, but also is interesting on its own
right.
