Multiplicative bases, Gröbner bases, and right Gröbner bases
This paper is the first on the theory of right Gröbner bases for algebras and modules. In section 2 the author defines a theory of a Gröbner basis for an algebra with a multiplicative basis which has an admissible order. (These concepts are defined in the paper.) What is a bit surprising is that the existence of such a basis implies that the algebra under consideration has to be a quotient of a path algebra \(K\Gamma\) by a special kind of ideal, called 2-nomial ideal, and the basis is the image of the paths under the epimorphism. This is also a nice justification for the introduction of path algebras. After that the author introduces the theory of right Gröbner bases of a right module over such kind of algebra. This can be applied immediately, and the author shows an application generalizing Cohn's theorem which in this case says that the path algebra is a hereditary ring. An algorithm to construct a tip reduced Gröbner basis is given and also the elimination theory. As a consequence there is an algorithm for find the intersection of right submodules of projective modules. This in its turn can be used to solve algorithmic linear systems of equations with coefficients in the algebra. There is a proof that if \(I\) is an ideal of the algebra such that the quotient is finite dimensional, (usually called an admissible ideal), then \(I\) has a finite reduced Gröbner basis. The paper is almost self contained and very readable, and a good introduction to the subject.
scientific article; zbMATH DE number 1471048