On Monic Gröbner Bases in Free Algebras over Rings Stemming from Bergman’s Diamond Lemma

Let R be an arbitrary commutative ring and R〈X〉 = R〈X1, ..., Xn〉 the free algebra of n generators over R. Note that Bergman’s diamond lemma characterizes the resolvability of ambiguities of monic relations of the form Wσ − fσ with fσ a linear combination of monomials ≺ Wσ, where ≺ is a semigroup partial ordering on 〈X〉; and that in the algorithmic language of Gröbner basis theory over a ground field K, the diamond lemma had been translated into the implementable termination theorem. Here we bring the termination theorem over R into play so that monic Gröbner Bases in R〈X〉 may be verified in an effective way, though it does not necessarily yield a noncommutative analogue of Buchberger Algorithm in such a setting. This enables us to recognize that many important algebras over rings may have Gröbner defining relations, and thereby enables us to study such algebras via their N-leading homogeneous algebra and BR-leading homogeneous algebra.