Zero divisors of Lie algebras

Let K be a field with characteristic zero. The definition of a zero divisor has been given for Lie algebras over the field K and with respect to this definition, we have given the definition of a domain, a semidomain and a strict semidomain for Lie algebras over the field K. We have given that the free amalgamated Lie products of two domains is a domain again. We have proven that a free metabelian Lie algebra and a finitely generated metabelian U - Lie algebra over the field K is a semidomain. Moreover, if A2 is the variety of all metabelian Lie algebras over the field K, we have given a criteria that establishes when metabelian product of two arbitrary nonzero metabelian Lie algebras A and B in the variety A2 is a semidomain. Finally, We have given criterias being a domain and a strict semidomain for semi simple Lie algebras.Key Words: Zero divisor, Domain, Semidomain, Strict semidomain, Semi simple