The structure of matrix polynomial algebras

Year 2023, , 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

References

• I. Assem, D. Simson and A. Skowronski, Elements of the Representation Theory of Associative Algebras, Volume 1 of London Mathematical Society Student Texts, Cambridge University Press, New York, 2006.
• M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Massachusetts, 1969.
• D. A. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics, Springer, 4th edition, 2015.
• D. Eisenbud, Commutative Algebra, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995.
• E. Eriksen, An Introduction to Noncommutative Deformations of Modules, In Noncommutative Algebra and Geometry, 243, 90-125, Lect. Notes Pure Appl. Math., Boca Raton, FL, Chapman & Hall Crc Edition, 2006.
• B. J. Gardner and R. Wiegandt, Radical Theory of Rings, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., 2004.
• T. W. Hungerford, Algebra, Graduate Texts in Mathematics, 73, Springer-Verlag, New York, 1974.
• N. Jacobson, Basic Algebra II, W. H. Freeman and Company, Second Edition, 1989.
• A. Kandri-Rody and W. Weispfenning, Noncommutative Gröbner bases in algebras of solvable type, J. Symbolic Comput., 9 (1990), 1-26.
• H. Kredel, Solvable Polynomial Rings, PhD thesis, Passau, 1992.
• O. A. Laudal, Noncommutative deformations of modules, Homology Homotopy and Applications, 4 (2002), 357-396.
• O. A. Laudal, Noncommutative algebraic geometry, Proceedings of the International Conference on Algebraic Geometry and Singularities (Spanish) (Sevilla, 2001). Rev. Mat. Iberoamericana, 19(2) (2003), 509-580.
• O. A. Laudal, Noncommutative Algebraic Geometry, In Proc. NATO Advanced Research Workshop on Computational Commutative and Noncommutative Algebraic Geometry, vol. 196 of NATO: Computer and Systems Sciences, 1-43. IOS Press, s. cojocaru et al. (eds) edition, 2005.
• L. Le Bruyn, Noncommutative Geometry and Cayley-Smooth Orders, Number 290 in Pure and Applied Mathematics. Boca Raton, FL 33487-2742, Chapman & Hall Crc Edition, 2008.
• T. Mora, Solving Polynomial Equation Systems, Cambridge University Press, Encyclopedia of Mathematics and its Applications Edition, 158, 2016.
• B. Nguefack and E. Pola, Effective Buchberger-Zacharias-Weispfenning theory of skew polynomial extensions of subbilateral coherent rings, J. Symbolic Comput., 99 (2020), 50-107.
• A. Siqveland, Geometry of noncommutative k-algebras, J. Gen. Lie Theory Appl., 5 (2011), Art. ID G110107 (12 pp).
• A. Siqveland, Introduction to non commutative algebraic geometry, J. Phys. Math., 133(6) (2015), Doi:10.4172/2090-0902.1000133.