Introduction into Noncommutative Algebra, Volume 1

TL;DR

This volume introduces fundamental concepts of noncommutative algebra over commutative rings, covering modules, algebras, linear maps, vector spaces, and eigenvalues in the context of associative division algebras.

Contribution

It provides a comprehensive foundational treatment of noncommutative algebra topics over commutative rings, including new insights into eigenvalues and linear systems in this setting.

Findings

Definitions of modules and algebras over commutative rings

Analysis of linear maps and eigenvalues over associative division algebras

Systematic study of linear equations in noncommutative algebra

Abstract

I dedicated the volume $1$ of monograph 'Introduction into Noncommutative Algebra' to studying of algebra over commutative ring. The main topics that I covered in this volume: definition of module and algebra over commutative ring; linear map of algebra over commutative ring; vector space over associative division algebra; system of linear equations over associative division algebra; homomorphism of vector space over associative division algebra and covariance theory; eigenvalue and eigenvector over associative division algebra.