On the radical of a free Malcev algebra

Abstract

Abstract. We prove that the prime radical rad M of the free Malcev algebra M of rank more than two over a field of characteristic = 2 coincides with the set of all universally Engelian elements of M. Moreover, let T (M) be the ideal of M consisting of all stable identities of the split simple 7-dimensional Malcev algebra M over F . It is proved that rad M = J(M) ∩ T (M), where J(M) is the Jacobian ideal of M. Similar results were proved by I. Shestakov and E. Zelmanov for free alternative and free Jordan algebras. An algebra M is called a Malcev algebra if it satisfies the identitieswhere J(x, y, z) = (xy)z + (zx)y + (yz)x is the Jacobian of the elements x, y, z [7,9,5]. Since for a Lie algebra the Jacobian of any three elements vanishes, Lie algebras fall into the variety of Malcev algebras. Among the non-Lie Malcev algebras, the traceless elements of the octonion algebra with the product given by the commutator [x, y] = xy − yx is one of the most important examples [9,5,6]. In 1977 I. P. Shestakov [11] proved that the free Malcev algebra M n on n ≥ 9 free generators is not semiprime; that is, M n contains nonzero nilpotent ideals. In 1979, V. T. Filippov [3] extended this result to free Malcev algebras with more than four generators. Therefore, the prime radical rad M n = 0 for n > 4, and a natural question on the description of this radical arises.For free alternative algebras, it was proved by Shestakov in [10] that the prime radical coincides with the set of nilpotent elements. A similar fact was established by E. Zel manov [15] for free Jordan algebras.In anticommutative algebras, the role of nilpotent elements is played by engelian elements. An element a of an algebra M is called engelian if the operator of right multiplication R a : x → xa is nilpotent. We will call an element a ∈ M universally engelian if, for every algebra M ⊇ M , the element a is engelian in M . In other words, the image R a of the element a in the (associative) universal multiplicative enveloping algebra R(M ) of M is nilpotent. In the present paper, we prove that the