|Abstract||In 1965, Buchberger introduced the notion of Gröbner bases for a polynomial ideal and an algorithm (Buchberger algorithm) for their computation; since the end of the seventies, Gröbner bases have been an essential tool in the development of computational techniques for the symbolic solution of polynomial systems of equations and in the development of effective methods in Algebraic Geometry and Commutative Algebra; moreover, Gröbner bases have been also generalized to free noncommutative algebra and to various noncommutative algebras, of interest in Differential Algebra (e.g. Weyl algebras, enveloping algebras of Lie algebras).|
The aim of this paper is to give an introduction, as elementary as I was able to make it, to both commutative and noncommutative algebras: Gröbner bases are in a sense a finite model of an infinite linear Gauss-reduced basis of an ideal viewed as a vector space and Buchberger algorithm is the corresponding generalization of the Gaussian elimination algorithm.
Moreover the paper contains a survey of some applications of Buchberger theory to noncommutative algebras; together with these results surveyed, this paper contains some minor new points: e.g. the "useless pair criteria" in the noncommutative case and the final result on the existence and "computability" of Gröbner bases for two-sided ideals in any finitely presented algebra.