On a theorem of Herstein

Abstract

Let A be a simple ring which is not a 4-dimensional algebra over a field of characteristic 2, and let U be a Lie ideal of A. Then Herstein proves in [2] that either U is contained in the center Z of A, or U contains [A, A ]; if in addition U is also assumed to be an associative ring and containls [A, A], then U=-A. Herstein and Baxter also prove in [1; 3], and [4] that [A, A] mod [A, A ]CnZ is a simple Lie ring. Our object here is to give a simple proof of this fact, basing our argument only on the results of [2] quoted above, though many of Herstein's ideas will be incorporated here without explicit mention. We first recall a few definitions and make some preliminary remarks. If u and a are elements of the ring A, [u, a] will designate uac-au, and [U, A ] will denote the module generated by all [u, a] where u is in U and a in A. U is an ideal of A if U is a module and if [U, A] is contained in U. If U and V are submodules of A then U+ V is the module generated by U and V. If U is a submodule of A we shall define U:A to be the set of elements x of A such that [x, A ] C U, thus [U: A, A ] is contained in U. It is easy to check that U:A is a ring from the identity [?v, a] = [u, va] + [v, au]. When U is a Lie ideal of A then U:A is also such. The following identities will be needed in the sequel; they are easy to verify: