New Conditions Establish the Relationships Between  Submodules

Khudhayer .O.  kadem

Authors

Khudhayer .O. kadem Ministry of Education- The Open Educational College-Babylon Study Center-Jableh Branch

Keywords:

Approximately Prime (App. Prime), Almost Approximately Nearly Prime (Al.App. Ne. Prime), Nearly Prime (Ne. Prime) Submodules

Abstract

This paper investigates new conditions that establish bidirectional relationships among several classes of submodules within the framework of commutative rings with identity and unitary modules. The study focuses on analyzing the structural connections between prime, nearly prime, approximately prime, and almost approximately nearly prime submodules by formulating precise criteria under which these concepts become equivalent. A series of theoretical results is developed to determine when the implications between these classes can be reversed, emphasizing the influence of module-theoretic properties such as the Jacobson radical, socle, essential submodules, and semisimplicity. Various propositions are proved to demonstrate that additional structural constraints on modules or submodules lead to equivalence between generalized notions and classical primeness. Illustrative examples are provided to clarify situations in which these relationships fail, highlighting the necessity of the imposed conditions. The obtained results contribute to a deeper understanding of generalized primeness in module theory and provide a unified perspective that connects several previously introduced generalizations through clearly defined algebraic conditions.

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