A Note on Nilpotent Jordan Rings

Let R be a 2-torsion free associative ring with involution. It is shown that if the set S of symmetric elements is nilpotent as a Jordan ring then R is nilpotent.

Some Radical Properties of Jordan Matrix Rings

Let A be a ring (not necessarily associative) in which 2x = a has a unique solution for each a ∈ A. Then it is known that if A contains an identity element 1 and an involution j : x ↦ x and if Ja is the canonical involution on An determined by where the ai al−l, 1 ≦ i ≦ n are symmetric elements in the nucleus of A then H(An, Ja), the set of symmetric elements of An, for n ≧ 3 is a Jordan ring if and only if either A is associative or n = 3 and A is an alternative ring whose symmetric elements lie in its nucleus [2, p. 127].

On Semiprime Ample Jordan Rings

The purpose of this paper is to point out that the arguments of [2] with slight modification extend the main result of [2] to the case of H satisfying either ACC or DCC on quadratic ideals and they extend [6, Theorem 2] to R being semiprime. Thus we obtainTheorem 1. Let R be a semiprime associative ring with involution ✶ and J a closed ample quadratic Jordan subring of H(R) satisfying either ACC or DCC on quadratic ideals. Then R is Goldie. In this case, J has a Jordan ring of quotients J′ which is a closed ample quadratic Jordan subring of H(R′) where R′ is the associative ring of quotients of R.

Certain Submodules of Simple Rings with Involution, II

Let R be a simple ring, of characteristic not 2, having an involution *. Let 5 = ﹛x G R|x* = x﹜ and K = ﹛x £ R|x* = — x﹜ be the set of symmetric and skew elements, respectively, of R.In [1] we discuss the structure of S as a Jordan ring and K as a Lie ring. In [2] we considered cross-over submodules, namely additive subgroups U ⊂ K, V ⊂ S such that