Generalized Malcev-Neumann Series Modules with the Beachy-Blair Condition

We introduce a new class of extension rings called the generalized Malcev-Neumann series ring R((S;σ;τ)) with coefficients in a ring R and exponents in a strictly ordered monoid S which extends the usual construction of Malcev-Neumann series rings. Ouyang et al. in 2014 introduced the modules with the Beachy-Blair condition as follows: A right R-module satisfies the right Beachy-Blair condition if each of its faithful submodules is cofaithful. In this paper, we study the relationship between the right Beachy-Blair condition of a right R-module MR and its Malcev-Neumann series module extension MSR((S;σ;τ)).