On the Square Submodule of a Mixed Module

The notion of the square submodule of a module M over an arbitrary com- mutative ring R; which is denoted by ⇔RM; was introduced by Aghdam and Najafizadeh in [3]. In fact, ⇔ sub R /sub M is the R-submodule of M generated by the images of all bilinear maps on M. Furthermore, given a submodule N of an R-module M; we say that M is nil modulo N if µ(MxM) ≤ N for all bilinear maps µ on M: The main question about the square submodule is that whether the quotient module M/⇔M is a nil module? In this paper, we investigate the square submodules of some classes of modules over commutative domains. Then, we have some results related to splitting modules which we need in our discussions. Finally, we give some examples of mixed Abelian groups A such that the quotient groups A/⇔A are not nil