The Radical of the Kernel of a Certain Differential Operator and Applications to Locally Algebraic Derivations

Abstract

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.