Prime differential nilalgebra exists

TL;DR

This paper constructs a monomorphism to prove the primality of certain differential algebras, solving a classical problem and providing new insights into their structure.

Contribution

It introduces a novel monomorphism from a differential algebra to a Grassmann algebra, establishing primality and solving Ritt's problem.

Findings

Proves the primality of $k\\{x\\}/[x^m]$ and its differential polynomial algebra.

Provides a new proof of the integrality of the ideal $[x^m]$.

Solves one of Ritt's longstanding problems.

Abstract

We construct a monomorphism from the differential algebra $k\{x\} / [x^m]$ to a Grassmann algebra endowed with a structure of differential algebra. Using this monomorphism we prove primality of $k\{x\} / [x^m]$ and its algebra of differential polynomials, solve one of Ritt's problems and give a new proof of integrality of the ideal $[x^m]$.