On algebras of rank three

An algebra of rank three is a commutative, finite dimensional algebra that may be defined by the property that every element generates a subalgebra of dimension not greater than two. In this article we discuss several classes of such algebras, including two classes related to central simple Jordan a lgebras, and derive some general results which indicate that, with the exce ption of one pathological class related to nilpotent algebras, every rank t hree algebra can be constructed either from a quadratic and alternative alg ebra or from a representation of a Clifford algebra. Among other results, s emisimple and simple rank three algebras are characterized, and the radical of an arbitrary rank three algebra is determined.