
Chapter 2
Noncommutative Associative
Algebras
The theory of commutative Gröbner bases is well known and properly docu-
mented in many textbooks, however it exhibits too many lucky coincidences
to be fully generalizable to more complex algebraic structures. The goal of this
chapter is to present the similar theory of noncommutative Gröbner bases in
a way that is amenable to adaptations for other algebraic structures discussed
in this book; as the reader will see later, in many ways, noncommutative Gröb-
ner bases are closer to Gröbner bases for operads than to Gröbner bases for
commutative associative algebras.
2.1 Introduction
In the classical case of