Multiplicative Bases, Gröbner Bases, and Right Gröbner Bases

In this paper, we study conditions on algebras with multiplicative bases so that there is a Gröbner basis theory. We introduce right Gröbner bases for a class of modules. We give an elimination theory and intersection theory for right submodules of projective modules in path algebras. Solutions to homogeneous systems of linear equations with coefficients in a quotient of a path algebra are studied via right Gröbner basis theory.