Abstract
In the first part of the paper, we provide a fairly complete description of a 2-torsion free simple ring R with involution f such that the skew traces relative to f of all elements of R form a commutative subset. In the remaining part, the results carried out earlier are put to work to delineate the structure of a simple Artinian ring R with involution f, which is equipped with a given non central submonoid M (= multiplicatively closed subset containing 1) preserved under f and under all inner automorphisms of R, and M is such that the skew traces of all elements of M form a commutative subset.
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