Dorroh extensions of algebras and coalgebras, I

TL;DR

This paper explores the structures and properties of Dorroh extensions in algebras and coalgebras, introduces finite duals for non-unital cases, and examines their dual relations.

Contribution

It provides a detailed description of Dorroh extensions and introduces finite duals for non-unital algebras and modules, establishing dual relations between algebraic and coalgebraic extensions.

Findings

Structures of Dorroh extensions are characterized.

Properties of these extensions are analyzed.

Dual relations between algebraic and coalgebraic extensions are established.

Abstract

In this article, we study Dorroh extensions of algebras and Dorroh extensions of coalgebras. Their structures are described. Some properties of these extensions are presented. We also introduce the finite duals of algebras and modules which are not necessarily unital. Using these finite duals, we determine the dual relations between the two kinds of extensions.