Centralizers in PBW Extensions

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.