Transitivity in QTAG-Modules : A study

Prof. Fahad  Sikander; Firdhousi Begum

doi:10.5269/bspm.68089

Prof. Fahad Sikander
Firdhousi Begum Aligarh Muslim University, Aligarh https://orcid.org/0000-0003-1441-3319

Abstract

In this paper, we explore the characteristics that ensure a $QTAG$-module $M$ is transitive or fully transitive, given that $H_\beta(M)$ is transitive or fully transitive for any ordinal $\beta$. We extend the concept of transitivity to include strong transitivity and projective full transitivity for QTAG-modules, and prove that a strongly transitive $QTAG$-module $M$ is fully transitive. We also demonstrate that the full transitivity of $M$ is equivalent to the full transitivity, transitivity, and projective full transitivity of $\bigoplus_{I} M$ for any index set $I$. We provide some examples and counterexamples to illustrate the concepts and results. Finally, we pose some open problems for further research, including the characterization of projectively fully transitive modules in terms of their endomorphism rings and the determination of the structure of projectively fully transitive modules over certain classes of rings.

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