Let R be a ring with unity and ∗ a ring involution on R. The set of ∗-symmetric elements, or just symmetric when the involution ∗ is clear from the context, is R∗ = {r ∈ R| r∗ = r}. We are going to denote by U(R) the group of units of R and by U∗(R) = {u ∈ U(R)| u∗ = u} the set of symmetric units. In general the set R∗ (respectively U∗(R)) is not a ring (resp. a subgroup). In fact R∗ (resp. U∗(R)) is a ring (resp. subgroup) if and only if the symmetric elements (resp. units) commute. Several papers have dealt with questions of how various algebraic prop-

erties of the set R∗ affect the structure of the whole ring. Similar question may be posed by making assumptions about the symmetric units or subgroup they generate.