Generalized Rigid Modules

Abstract

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 α-rigid if maα(a) = 0 implies ma = 0 for any m ∈ M and a ∈ R. We investigate some properties of αrigid 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; α] is semicommutative if and only if M [x; α] R[[x; α]] is semicommutative.

Generalized rigid modules and their polynomial extensions

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Let R be a ring with unity, σ an endomorphism of R and M R a right Rmodule. In this paper, we continue studding σ-rigid modules that were introduced by Gunner et al [13]. We give some results on σ-rigid modules and related concepts.

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“…Motivated by the properties of σ-rigid rings that have been studied in [10,11,12,14], Guner el al. [13], introduced σ-rigid modules as a generalization of σ-rigid rings. A module M R is called σ-rigid, if maσ(a) = 0 implies ma = 0 for any m ∈ M and a ∈ R. Clearly, σ-reduced modules are σ-rigid, but the converse need not be true [13,Example 2.18].…”

“…[13], introduced σ-rigid modules as a generalization of σ-rigid rings. A module M R is called σ-rigid, if maσ(a) = 0 implies ma = 0 for any m ∈ M and a ∈ R. Clearly, σ-reduced modules are σ-rigid, but the converse need not be true [13,Example 2.18]. Recall that, a module M R is σ-reduced if and only if it is σ-semicommutative and σ-rigid [13,Theorem 2.16].…”

“…A module M R is called σ-rigid, if maσ(a) = 0 implies ma = 0 for any m ∈ M and a ∈ R. Clearly, σ-reduced modules are σ-rigid, but the converse need not be true [13,Example 2.18]. Recall that, a module M R is σ-reduced if and only if it is σ-semicommutative and σ-rigid [13,Theorem 2.16]. Thus, the concept of σ-reduced modules is connected to the σ-semicommutative and σ-rigid modules.…”

“…y be the ring of polynomials in two non-commuting indeterminates over a field K. Consider the right R-module M = R/xR. The module M is rigid but not semicommutative by[13, Example 2.18].…”