Bergman's diamond lemma

In mathematics, specifically the field of abstract algebra, Bergman's Diamond Lemma (after George Bergman) is a method for confirming whether a given set of monomials of an algebra forms a ${\displaystyle k}$-basis. It is an extension of Gröbner bases to non-commutative rings. The proof of the lemma gives rise to an algorithm for obtaining a non-commutative Gröbner basis of the algebra from its defining relations. However, in contrast to Buchberger's algorithm, in the non-commutative case, this algorithm may not terminate.