TL;DR
This paper introduces the concept of strongly prime submodules in modules over commutative rings, extending properties of prime ideals and generalizing the Principal Ideal Theorem to modules.
Contribution
It defines strongly prime submodules and demonstrates that they retain key properties of prime ideals, including an extension of the Principal Ideal Theorem to modules.
Findings
Strongly prime submodules inherit essential properties of prime ideals.
The Generalized Principal Ideal Theorem is extended to modules.
The notion provides a new perspective on submodule structure.
Abstract
Let be a commutative ring with identity. For an -module , the notion of strongly prime submodule of is defined. It is shown that this notion of prime submodule inherits most of the essential properties of the usual notion of prime ideal. In particular, the Generalized Principal Ideal Theorem is extended to modules.
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Taxonomy
TopicsRings, Modules, and Algebras · Advanced Topics in Algebra · Commutative Algebra and Its Applications