  In this article we give a description for the centralizer of the coefficient
ring $R$ in the skew PBW extension $\sigma(R)<x_1,x_2,\cdots,x_n>.$ We give an
explicit description in the quasi-commutative case and state a necessary
condition in the general case. We also consider the PBW extension
$\sigma(\mathcal{A})<x_1,x_2,\cdots,x_n>$ of the algebra of functions with
finite support on a countable set, describing the centralizer of $\mathcal{A}$
and the center of the skew PBW extension.
