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In this paper we consider algebras satisfying the identity (I) x(xy) +(yx)x =2(xy)x and show that there are no nodal algebras of this type over any field F of characteristic zero. The proof is obtained by first showing that if x is an element of a finitedimensional algebra satisfying (I) over a field of characteristic zero then the operator L(x) -R(x) is nilpotent. A finite-dimensional power-associative algebra A with identity 1 over a field F is called a nodal algebra [7 ] if every x in A is of the form x =al +n where a is in F and n is nilpotent, and the set N of nilpotent elements of A does not form a subalgebra of A. Albert [1I has proved that there are no commutative nodal algebras over any field F of characteristic zero by showing that N forms a subalgebra. There do exist, however, examples of nodal algebras over fields of characteristic zero [2 ]. Algebras satisfying (I) have been studied by Kosier [5], Witthoft [8] and the author [6]. It should be noted that in linearized form (I) reduces to (1) x(zy) + z(xy) + (yx)z + (yz)x = 2(xy)z + 2(zy)x and in operator form (I) is just (2) L(x)2 + R(x)2 = 2L(x)R(x) or (3) L(x)(L(x) R(x)) = (L(x) R(x))R(x) where L(x)(R(x)) is the operator which acts as follows: yL(x) =xy(yR(x) = yx). In a commutative algebra L(x) -R(x) = 0. For algebras satisfying (I) we have the following. THEOREM 1. Let A be a finite-dimensional algebra satisfying (I) over a field F of characteristic zero. Then for any element x the operator L(x) -R(x) is nilpotent. Received by the editors June 27, 1969 and, in revised form, September 15, 1969. A MS Subject Classifications. Primary 1720.

On nearly commutative nodal algebras in characteristic zero