Zip Property on Malcev-Neumann Series Modules

Abstract

Let R be a ring, MR a right R-module, G a totally ordered group, σ a map from G into the group of automorphisms of R which assigns to each x ∈ G an automorphism σ_x ∈ Aut(R), τ a map from G × G to U(R) (the group of unit elements of R) and M((G; σ ; τ)) the Malcev-Neumann series module. Then, under some certain conditions, we show that MR is a right zip R-module if and only if M((G; σ ; τ))_{R((G;σ ;τ))} is a right zip R((G; σ ; τ))-module, where R((G; σ ; τ)) is the Malcev-Neumann series ring.