A LITTLE WEDDERBURN PRINCIPAL THEOREM

We point out that, even if A/rad A is inseparable, the finite-dimensional algebra (or, more generally, left Artinian ring) A admits a canonical ideal direct sum decomposition A = A0 + A# such that rad A0 = (0), rad A# = rad A, rad A# is an essential ideal of A#, and A# is unital if and only if A is unital. This is a consequence of a general and maximally elementary splitting of an arbitrary ring relative to a suitable nil ideal. This process of essentializing a nil ideal is useful in the study of categories of ideal and radical embeddings.