Abstract

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)).

Highlights

  • Throughout this paper R denotes an associative ring with identity; (S, ⋅, ⩽) is a strictly ordered monoid (i.e., (S, ⩽) is an ordered monoid satisfying the conditions that if s < s󸀠, st < s󸀠t and ts < ts󸀠 for s, s󸀠, t ∈ S)

  • 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

  • We study the relationship between the right Beachy-Blair condition of a right R-module MR and its Malcev-Neumann series module extension M((S))R((S;σ;τ))

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Summary

Introduction

Throughout this paper R denotes an associative ring with identity; (S, ⋅, ⩽) is a strictly ordered monoid (i.e., (S, ⩽) is an ordered monoid satisfying the conditions that if s < s󸀠, st < s󸀠t and ts < ts󸀠 for s, s󸀠, t ∈ S). We study the relationship between the right Beachy-Blair condition of a right R-module MR and its Malcev-Neumann series module extension M((S))R((S;σ;τ)). If MR is a unitary right R-module, the MalcevNeumann series module B = M((S)) is the set of all formal sums ∑x∈S mxx with coefficients in M and artinian and narrow supports, with pointwise addition and scalar multiplication rule is defined by

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