Abstract. In this paper we prove that the subalgebras of cocommutative elements in the quantized coordinate rings of Mn, GLn and SLn are the centralizers of the trace x 1,1 + · · · + xn,n in each algebra, for q ∈ C × being not a root of unity. In particular, it is not only a commutative subalgebra as it was known before, but it is a maximal one.