Weakly 2-quasi-prime sub-modules

Let R be a commutative ring with identity, and let M be a left R-module. We define a proper sub-module N of an R-module M to be a weakly 2-quasi-prime sub-module if whenever 0≠r_1 r_2 m∈N, r_1 r_2∈R, m∈M, then either 〖r_1〗^2 m∈N or ∈〖r_2〗^2 m∈N. This concept is an expansion of the idea of a 2-quasi-prime sub-module, where a proper sub-module N of an R-module M is said to be a 2-quasi-prime sub-module if for all a,b∈R, x∈M and abx∈N then either a^2 x∈N or〖 b〗^2 x∈N. Various properties of weakly 2-quasi-prime sub-modules are considered.