A derivation on an algebra which is not inner.

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Suppose that A is an algebra with unit 1 and a is an element of A which is not algebraic (i.e. {1, a, a^{2}, ⋯} isn’t a linearly independent set). Let B be the subalgebra of A generated by 1 and a. Define a mapping D of B into B by D(λ_{0} + λ_{1}a + ⋯ + λ_{n}a^{n}) = λ_{1} + 2λ_{2}a + ⋯ + nλ_{n}a^{n−1}. Obviously D is a derivation on B and isn’t inner, since B is commutative and D ≠ 0.