On the radical of a free Malcev algebra

Abstract

We prove that the prime radical radM 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 radM = 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 identities x = 0, J(x, xy, z) = J(x, y, z)x, where 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 Mn on n ≥ 9 free generators is not semiprime; that is, Mn 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 radMn = 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 Ra : 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 Ra 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 Received by the editors February 23, 2011 and, in revised form, March 31, 2011. 2010 Mathematics Subject Classification. Primary 17D10, 17D05, 17A50, 17A65.