Linear algebras in which division is always uniquely possible

does not vanish. Hence the condition that right hand [left baiid] division shall always be uniquely possible is that A,. [Al] shall vanish only when every a. vanishes. Now either of these conditions is satisfied when the other is, since either is equivalent to the condition that a product shall vanish only when one factor vanishes. We consider algebras in which these conditions are satisfied and in which there is a modulus, i. e., an element 1 such that 1A = Al = A for every element A. We shall henceforth set el = 1. For m = 2, the algebra is the field F(e2). Indeed, e2 _ e2y222 0 is irreducible in F since Ar al + al a2 y222 a a2 y221. In ? 2 I consider the general transformation of algebras with three units, exhibiting families of algebras invariant under every linear transformation and determining the algebras which admit more than one transforination into itself (and hence exactly three transformations). Froin each standpoint I am led to the same remarkable set of families of algebras, each set characterized by a parameter ,u. For , = 1, the family consists of all fields of rank three with respect to F. For u = 0, the commutative algebras have the property that division is always possible.