Type: Article
Publication Date: 1975-10-01
Citations: 35
DOI: https://doi.org/10.2140/pjm.1975.60.131
We give a systematic account on the relationship between a ring R with involution and its subrings S and K, which are generated by all its symmetric elements or skew elements respectively. PJEK-HWEE LEERings with involution abound with examples of Lie ideals.One can easily show that any subring, generated by symmetric elements and containing T = {x + x*|JC G R} the set of all traces, must be a Lie ideal.In particular, both S and T are Lie ideals.Another essential property of S follows from the next lemma.We denote by N the set of all norms, i.e.N = {xx*| JC G R}. LEMMA 2. Let U be an additive subgroup of R such that T C U C S andxUx*C Ufor all x G R. IfN C 17, thenxUx*Q U for all x G R.Proof.We prove by induction that xu x u n x * G Ό for all x G R and Mi, , u n G (7.The case n = 1 is clear.Assume the assertion holds for n -1 thenbecause C7 is a Lie ideal.DEFINITION.A subring U of R is called a symmetric subring if: 1. U is generated by a set of symmetric elements.2. TUNCU 3. xUx*QU for all x <Ξ R.LEMMA 56.Let R be a *-prime ring and e the identity of K or V 2 .If e ^ 0, then it is the identity of R.