By Thomas Muir, revised by W.H. Metzler

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This contradiction completes the proof of the present implication. (iv) implies (i): For such a representation of an element f , we have lm(f ) = lm(mi gi mi ) = mi lm(gi )mi , for some i, so lm(f ) is divisible by lm(gi ). Since this is assumed true for every f ∈ I, it follows that G is a Gröbner basis. 6 leads naturally to a recipe for computing reduced Gröbner bases: given a set of generators of an ideal, one has to compute all pairwise S-polynomials, adjoin all reduced forms of those to the set of generators, and repeat the same.

Suppose that lm(gk−1 ) and lm(gk ) have an overlap inside m. Without loss of generality, we have lm(gk−1 ) = u1 v and lm(gk ) = vu2 for some monomials u1 , u2 , v with v = 1, so that mk−1 u1 vmk−1 = mk−1 lm(gk−1 )mk−1 = m = mk lm(gk )mk = mk vu2 mk . This implies that mk = mk−1 u1 , and mk−1 = u2 mk . Recall the definition of the corresponding S-polynomial Sv (gk−1 , gk ) = gk−1 u2 − u1 gk , which we will use in the form u1 gk = gk−1 u2 − Sv (gk−1 , gk ). Noncommutative Associative Algebras 35 Let us examine the sum ck−1 mk−1 gk−1 mk−1 + ck mk gk mk : ck−1 mk−1 gk−1 u2 mk + ck mk−1 u1 gk mk = ck−1 mk−1 gk−1 u2 mk + ck mk−1 (gk−1 u2 − Sv (gk−1 , gk ))mk = (ck−1 + ck )mk−1 gk−1 u2 mk − ck mk−1 Sv (gk−1 , gk )mk .

Sm . 14 Algebraic Operads: An Algorithmic Companion Recall that the row canonical form of a matrix A, is a matrix R obtained from A by elementary row operations for which the first nonzero entry of each nonzero row of R is equal to 1 (this entry is called the pivot of that row), the positions of the pivots increase with the increase in the row number, and all entries in each column containing a pivot are equal to zero. The following algorithm for computing canonical forms of matrices is wellknown.