Purely Inseparable ring extensions

TL;DR

This paper explores purely inseparable ring extensions, providing new characterizations using differential operators, analyzing intermediate rings, and extending Galois correspondence concepts to commutative rings.

Contribution

It introduces differential operator-based characterizations, generalizes Galois correspondence, and investigates properties of intermediate and composite extensions in purely inseparable ring extensions.

Findings

Characterization of purely inseparable extensions via differential operators

Generalization of Jacobson-Bourbaki Galois correspondence to rings

Conditions under which inseparability propagates in towers

Abstract

We revisit the concept of special algebras, also known as \textit{purely inseparable ring extensions}. This concept extends the notion of purely inseparable field extensions to the more general context of extensions of commutative rings. We use differential operators methods to provide a characterization for a ring extension to be purely inseparable in terms of a condition on certain modules of differential operators associated to the ring extension. This approach is also used to recover an already known characterization involving the modules of principal parts. Next, given a purely inseparable ring extension $A\subset C$, we aim to understand which intermediate rings $A\subset B\subset C$ satisfy the property that both $A\subset B$ and $B\subset C$ are both flat extensions by considering only the subalgebra $\operatorname{End}_B(C)$ of $\operatorname{End}_A(C)$. To achieve this, we prove a generalization of the Jacobson-Bourbaki theorem on Galois correspondence for field extensions to the setting of commutative ring extensions with homeomorphic spectra. Finally, given a tower of ring extensions $A\subset B\subset C$, we consider the question of whether the fact that two of the three extensions $A\subset C$, $A\subset B$, and $B\subset C$ are purely inseparable implies that the third one is also purely inseparable.