  In this paper we prove that the subalgebras of cocommutative elements in the
quantized coordinate rings of $M_{n}$, $GL_{n}$ and $SL_{n}$ are the
centralizers of the trace $x_{1,1}+\dots+x_{n,n}$ in each algebra, for
$q\in\mathbb{C}^{\times}$ 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.
