On some H-cleft extensions which are distinguished by their polynomial H-identities

TL;DR

This paper investigates H-cleft extensions over a finite commutative ring R with Taft algebra H, demonstrating that when N is a unit, these extensions are uniquely characterized by their polynomial H-identities.

Contribution

It establishes that all H-cleft extensions over R are classified by polynomial H-identities when N is a unit, providing a new understanding of their structure.

Findings

H-cleft extensions are determined by polynomial H-identities

Classification of extensions is complete when N is a unit

Extension isomorphism is characterized by polynomial identities

Abstract

Let H be the Taft algebra over a finite commutative ring R. When N is a unit in R, we show that all H-cleft extensions over R are determined up to H-comodule algebra isomorphism by their polynomial H-identities.