On finite dimensional algebras with only trivial derivations(automorphisms) and simple algebras

Abstract: This paper deals with $n$-dimensional algebras, over any field, which have only trivial derivation (automorphism) and simple algebras. It is shown that the corresponding sets of algebras are not empty and, in algebraically closed field case, they are dense subsets of the variety of $n$-dimensional algebras with respect to the Zariski topology. Moreover, an inductive construction method is offered to create these kind of algebras as well. In two-dimensional case a complete classifications, up to isomorphism, of such algebras are provided.