On n-Trivial Extensions of Rings

TL;DR

This paper generalizes the classical trivial extension of rings to n-trivial extensions involving multiple modules, exploring their properties, graded structures, and divisibility aspects, thus broadening the understanding of ring extensions.

Contribution

It introduces the concept of n-trivial extensions of rings, extending classical results and providing new insights into their properties and structures.

Findings

Generalization of trivial extensions to n-trivial extensions

Analysis of graded structures in n-trivial extensions

Investigation of divisibility properties and open questions

Abstract

The notion of trivial extension of a ring by a module has been extensively studied and used in ring theory as well as in various other areas of research like cohomology theory, representation theory, category theory and homological algebra. In this paper we extend this classical ring construction by associating a ring to a ring $R$ and a family $M=(M_i)_{i=1}^{n}$ of $n$ $R$-modules for a given integer $n\geq 1$. We call this new ring construction an $n$-trivial extension of $R$ by $M$. In particular, the classical trivial extension will be just the $1$-trivial extension. Thus we generalize several known results on the classical trivial extension to the setting of $n$-trivial extensions and we give some new ones. Various ring-theoretic constructions and properties of $n$-trivial extensions are studied and a detailed investigation of the graded aspect of $n$-trivial extensions is also given. We end the paper with an investigation of various divisibily properties of $n$-trivial extensions. In this context several open questions arise.