The structure of matrix polynomial algebras

Authors : Bertrand NGUEFACK

Pages : 137-177

Doi:10.24330/ieja.1151001

View : 19 | Download : 11

Publication Date : 2023-01-09

Article Type : Research Paper

Abstract :This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions of a coefficient algebra by insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(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