  In this paper, we consider those multiplication operators M_p on the Bergman
space L_a^2(D^2) over the bidisk, defined by a class of polynomials p. Also,
this paper consider the reducing subspaces of M_p, the von Neumann algebra
W^*(p) generated by M_p, and its commutant V^*(p)=W^*(p)'. The structure of
V^*(p) is completely determined, along with those reducing subspaces of M_p.
