Wedderburn's best-known paper was his sole-authored "On hypercomplex numbers," published in the 1907 Proceedings of the London Mathematical Society, and for which he was awarded the D.Sc. the following year. This paper gives a complete classification of simple and semisimple algebras. He then showed that every semisimple algebra finite-dimensional can be constructed as a direct sum of simple algebras and that every simple algebra is isomorphic to a matrix algebra for some division ring. The Artin–Wedderburn theorem generalises this result, with the ascending chain condition.