NAZIM AGAYEV, SAIT HALICIOĞLU and ABDULLAH HARMANCI
On symmetric modules
Received: 18 March 2009
Accepted in revised form: 19 May 2009
Mathematics Subject Classification (2000): 16U80
Abstract Let α be an endomorphism of an arbitrary ring R with identity and let M be a right R-module. We introduce the notion of α-symmetric modules as a generalization of α-reduced modules. A module M is called α-symmetric if, for any m ∈ M and any a, b ∈ R, mab = 0 implies mba = 0; ma = 0 if and only if mα(a) = 0. We show that the class of α-symmetric modules lies strictly between classes of α-reduced modules and α-semicommutative modules. We study characterizations of α-symmetric modules and their related properties including module extensions. For a rigid module M, M is α-reduced if and only if M is α-symmetric. For a module M, it is proved that M[x]R[x] is α-symmetric if and only if M[x, x -1]R[x, x -1] is α-symmetric.