On $n$-trivial extensions of rings

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 such as 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 merely be 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 finish the paper with an investigation of various divisibility properties of $n$-trivial extensions. In this context, several open questions arise.