Algebras Defined by Monic Gr\"obner Bases over Rings

TL;DR

This paper demonstrates that monic Gr"obner bases over a field can be extended to arbitrary rings, enabling the study of R-algebras with such bases using PBW structure theory similar to that over fields.

Contribution

It establishes a correspondence between monic Gr"obner bases over fields and rings, facilitating the analysis of R-algebras with these bases through PBW theory.

Findings

Monic Gr"obner bases over fields extend to rings.

Many R-algebras have defining relations forming monic Gr"obner bases.

Enables PBW structure analysis for R-algebras with such bases.

Abstract

Let $K\langle X\rangle =K\langle X_1,...,X_n\rangle$ be the free algebra of $n$ generators over a field $K$, and let $R\langle X\rangle =R\langle X_1,...,X_n\rangle$ be the free algebra of $n$ generators over an arbitrary commutative ring $R$. In this semi-expository paper, it is clarified that any monic Gr\"obner basis in $K\langle X\rangle$ may give rise to a monic Gr\"obner basis of the same type in $R\langle X\rangle$, and vice versa. This fact turns out that many important $R$-algebras have defining relations which form a monic Gr\"obner basis, and consequently, such $R$-algebras may be studied via a nice PBW structure theory as that developed for quotient algebras of $K\langle X\rangle$ in ([LWZ], [Li2, 3]).