The structure of matrix polynomial algebras

Year 2023, Volume: 33 Issue: 33, 137 - 177, 09.01.2023

Bertrand Nguefack

https://doi.org/10.24330/ieja.1151001

Cited By: 1

Abstract

This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions
of a coefficient algebra by (elementary) matrix-variables over
a ground polynomial ring in not necessary commuting variables.
These matrix subalgebras of full matrix rings over polynomial rings show up
in noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining a precise lift of the Hilbert Basis Theorem when the
ground ring is either a commutative polynomial ring, a free noncommutative polynomial ring or a skew polynomial ring extension by a free commutative term-ordered monoid.
We equally address the natural but rather delicate question of recognising which matrix polynomial algebras are Cayley-Hamilton algebras,
which are interesting noncommutative algebras arising from the study of $\mathrm{Gl}_{n}$-varieties.

Keywords

Matrix algebra, polynomial ring, noncommutative algebraic geometry

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