Non-commutative reduction rings

Abstract

Reduction relations are means to express congruences on rings. In the special case of congruences induced by ideals in commutative polynomial rings, the powerful tool of Grabner bases can be characterized by properties of reduction relations associated with ideal bases. Hence, reduction rings can be seen as rings with reduction relations associated to subsets of the ring such that every finitely generated ideal has a finite Gröbner basis. This paper gives an axiomatic framework for studying reduction rings including non-commutative rings and explores when and how the property of being a reduction ring is preserved by standard ring constructions such as quotients and sums of reduction rings, as well as extensions to polynomial and monoid rings over reduction rings. Moreover, it is outlined when such reduction rings are effective