In this paper we give some relationships among the Gröbner-Shirshov bases in free associative algebras, free left modules and "double-free" left modules (free modules over a free algebra). We give the Chibrikov's Composition-Diamond lemma for modules and show that Kang-Lee's Composition-Diamond lemma follows from this lemma. As applications, we also deal with highest weight module over the Lie algebra sl 2 , Verma module over a Kac-Moody algebra, Verma module over Lie algebra of coefficients of a free conformal algebra and the universal enveloping module for a Sabinin algebra.

In this paper, we generalize the Shirshov's Composition Lemma by replacing the monomial order for others. By using Gröbner-Shirshov bases, the normal forms of HNN extension of a group and the alternating group are obtained.

In this paper, we give a Gröbner-Shirshov basis of the braid group B n+1 in Adyan-Thurston generators. We also deal with the braid group of type B n . As results, we obtain a new algorithm for getting the Adyan-Thurston normal form, and a new proof that the braid semigroup B + n+1 is the subsemigroup in B n+1 .

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In this paper, we prove that two-generator one-relator groups with depth less than or equal to 3 can be effectively embedded into a tower of HNN-extensions in which each group has the effective standard normal form. We give an example to show how to deal with some general cases for one-relator groups. By using the Magnus' method and Composition-Diamond lemma, we reprove the G. Higman, B. H. Neumann and H. Neumann's embedding theorem.In this section, we will cite some literatures about the definition of groups with the standard normal form and Composition-Diamond lemma on free associative algebra k X .Then H is called an HNN-extension of G relative to A, B and φ.Then H is called a group with the stable letter t and the base group G.Generally, we may use groups with (many) stable letters T = {t}. Remark: Let H be in Definition 2.2. P. S. Novikov ([18, 19, 20]) called the letter t to be regular if the subgroup gp A i |i ∈ I , gp B i |i ∈ I of G are isomorphic by ϕ : A i → B i , i ∈ I. Thus, Novikov's group G with a regular stable letter t and the base G is exactly an HNN-extension of G.Define the corresponding words relative to a stable letter t by the above relations:Moreover, for convenience, we put A(t −1 ) = B(t) and B(t −1 ) = A(t). Then, it is clear that for anyLet G 0 → G 1 → · · · → G n be a tower of groups, where G i+1 is a group with some stable letters and the base group G i for each i. We call such a tower a Novikov tower. Moreover, if each G i+1 is an HNN-extension of G i , then we call this tower a tower of HNN-extensions or B-tower (Britton tower, see [7]).Let G 0 → G 1 → · · · → G n be a Novikov tower with G 0 free. If p is a stable letter of G i+1 , we say the weight of p to be i+1. Taking an arbitrary relation Ap = pB (A, B ∈ G i ) from G i+1 , we can represent it as follows:where x, y are some stable letters of the highest weight in the words A and B, respectively. We call x, y to be distinguishing letters of the relation Ap = pB. We associate four types of forbidden subwords in G i+1 (see [6], §6.4):xB(x)A ′′ p, x −1 B(x −1 )A ′−1 p, yB(y)B ′′ p −1 , y −1 B(y −1 )B ′−1 p −1Define the set C i of words in G i (0 ≤ i ≤ n) as follows (see [6,11,12]).