Multiplication modules and homogeneous idealization III

In our recent work we gave a treatment of certain aspects of multiplication modules, projective modules, flat modules and cancellation-like modules via idealization. The purpose of this work is to continue our study and develop the tool of idealization, particularly in the context of closed, divisible injective, and simple modules. We determine when a ring R (M), the idealization of M, is a quasi-Frobenius or a distinguished ring. We also introduce and investigate the concept of M-1/2 (weak) cancellation ideals.