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.

Some minor improvements. Latex 22 pages

References in corpus (2)

• Images of Ideals under Derivations and $\mathcal{E}$-Derivations of Univariate Polynomial Algebras over a Field of Characteristic Zero
• The Gaussian Moments Conjecture and the Jacobian Conjecture