Locally Finite Associative Algebras and Their Lie Subalgebras

An infinite dimensional associative algebra A over a field F is called locally finite associative algebra if every finite set of elements is contained in a finite dimensional subalgebra of A . Given any associative algebra A over field F of any characteristic. Consider a new multiplication on A called the Lie multiplication which defined by [ a, b ] = ab − ba for all a, b ∈ A , where ab is the associative multiplication in A . Then L = A (−) together with the Lie multiplication form a Lie subalgebra of A . It is natural to expect that the structures of L and A are connected closely. In this paper, we study and discuss the structure of infinite dimensional locally finite Lie and associative algebras. The relation between them, their ideals and their inner ideals is considered. A brief discussion of the simple associative algebras and simple Lie algebras is also be provided.