Radical-based generalizations of $H$-supplemented modules

Articles in Press

Document Type : Research Paper

Authors

1 Middle Technical University, Technical College of Management, Baghdad, Iraq.

2 College of Education for Pure Science Ibn Al-Haitham, Department of Mathematics, Baghdad, Iraq.

3 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.

Abstract

In this paper, we introduce a novel extension of the well-studied class of lifting modules, which we call $J$-$H$-supplemented modules. This new class is closely tied to the concept of $J$-small submodules, a refinement of small submodules characterized by their interaction with the Jacobson radical. A key feature of $J$-$H$-supplemented modules is the existence, for every submodule $U$ of $W$ that satisfies a certain radical condition on the quotient $W/U$, of a direct summand $T$ of $W$ that reflects the relative position of $U$ with respect to other submodules. More precisely, these modules are defined by the property that for every submodule $U$ and for all submodules $P$ where $U+P=W$, the equality $T+P=W$ also holds if and only if $T$ is a direct summand of $W$ and the quotient $W/U$ is a radical module.
This characterization establishes a deep connection between $J$-$H$-supplemented modules and the structure of radicals in quotient modules, thereby extending the classical theory of lifting modules by incorporating radical-sensitive conditions. Our results demonstrate how the presence of such direct summands governs the decomposition properties of these modules and sheds light on new structural phenomena in module theory. This framework not only broadens the theoretical foundation of supplemented modules but also opens pathways for further exploration of module decompositions influenced by radical submodules.

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References

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Algebraic Structures and Their Applications

Articles in Press, Corrected Proof
Available Online from 23 December 2025

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