Let H2(D2) be the Hardy space over the bidisk D2, and let M? = [(z??(w))2] be the submodule generated by (z??(w))2, where?(w) is a function in H∞(w). The related quotient module is denoted by N? = H2(D2)-M? . In the present paper, we study the Fredholmness of compression operators Sz, Sw on N? . When?(w) is a nonconstant inner function, we prove that the Beurling type theorem holds for the fringe operator Fw on [(z?w)2]-z[(z?w)2] and the Beurling type theorem holds for the fringe operator Fz on M? -wM? if?(0)=0. Lastly, we study some properties of Fw on [(z?w2)2]-z[(z?w2)2].