Abstract

GUNER, ERDAL  and  HALICIOGLU, SAIT. Generalized Rigid Modules. Rev.colomb.mat. [online]. 2014, vol.48, n.1, pp.111-123. ISSN 0034-7426.  https://doi.org/10.15446/recolma.v48n1.45198.

Let α be an endomorphism of an arbitrary ring R with identity. The aim of this paper is to introduce the notion of an α-rigid module which is an extension of the rigid property in rings and the α-reduced property in modules defined in [8]. The class of α-rigid modules is a new kind of modules which behave like rigid rings. A right R-module M is called \alpha-rigid if ma α(a)=0 implies ma=0 for any m ∈ M and a ∈ R. We investigate some properties of α-rigid modules and among others we also prove that if M[x;α] is a reduced right R[x;α]-module, then M is an α-rigid right R-module. The ring R is α-rigid if and only if every flat right R-module is α-rigid. For a rigid right R-module M, M is α-semicommutative if and only if M[x;α]_{R[x;\,\alpha]} is semicommutative if and only if M\big[[x;α]\big]_{R[[x;\,\alpha]]} is semicommutative.

Keywords : Reduced modules; Semicommutative modules; Armendariz modules; Rigid modules.

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