Abstract. Let θ : M → N be a zero-product preserving linear map between algebras. We show that under some mild conditions θ is a product of a central element and an algebra homomorphism. Our result applies to matrix algebras, standard operator algebras, C * -algebras and W * -algebras.

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In this paper, we study the concept of associative n-conformal algebra over a field of characteristic 0 and establish Composition-Diamond lemma for a free associative n-conformal algebra. As an application, we construct Gröbner-Shirshov bases for Lie n-conformal algebras presented by generators and defining relations.

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We found Gröbner-Shirshov basis for the braid semigroup B + n+1 . It gives a new algorithm for the solution of the word problem for the braid semigroup and so for the braid group.