On J-torsionless modules
Dimpy Mala Dutta, A. M. Buhphang, M. B. Rege
TL;DR
The paper introduces the JReject of a class of modules and the related notion of J-torsionless modules, framing them as a generalization of the classical reject and connecting cogeneration by $R/J(R)$. It develops the foundational machinery via $JRej_M(\mathscr{U})$, relates it to $Rej_M(\mathscr{U})$, and examines its behavior under morphisms and direct sums, with $JRej_R(\mathscr{U})$ forming a two-sided ideal. J-torsionless modules are characterized equivalently as cogenerated by $R/J(R)$ and as submodules of products of $R/J(R)$, with broad closure properties and links to regular, W-regular, and fully idempotent notions. The results yield structural insights for rings, including conditions under which all simple modules are J-torsionless, and identify consequences for LA-rings, V-rings, and self-injective rings, supported by illustrative examples.
Abstract
In this paper, we introduce the concept of JReject of a class of modules as a generalization of the notion of reject of a class of modules. We also introduce the notion of J-torsionless modules and give a characterization of regularity on the basis of the J-torsionless condition. A necessary and sufficient condition on $R$ is also given for every cyclic module over $R$ to be J-torsionless. Finally, we give a description of self-injective rings over which every module is J-torsionless.