From cohomology theory we know that separables algebras $A$ possess only inner derivations. In fact, they possess only inner Derivation into every $A$-bimodule $M$ which characterize separable algebras.
For example, the Algebra of upper triangular matrices over a field or the Solomon-Algebra in characteristic Zero possess only inner derivations.
My question is to describe those associative algebras -- maybe also finite-dimensional -- which possess only inner derivations.
