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The unique maximal GF-regular submodule of a module.

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  • 1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China.

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Summary
This summary is machine-generated.

This study introduces GF-regular modules in R-module theory. We proved that every unitary R-module has a unique maximal GF-regular submodule, denoted M GF(A), and explored its radical properties.

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Area of Science:

  • Abstract Algebra
  • Module Theory
  • Ring Theory

Background:

  • Introduces the concept of GF-regular modules within the framework of R-module theory.
  • Defines a GF-regular module A (over a ring R) where for every element a in A and r in R, there exist t in R and a positive integer n such that r(n)tr(n)a = r(n)a.

Purpose of the Study:

  • To establish the existence and uniqueness of a maximal GF-regular submodule within any unitary R-module.
  • To investigate the radical properties associated with GF-regular submodules.
  • To explore the relationship between GF-regular submodules and other module concepts like projectivity and G-purity.

Main Methods:

  • The study employs fundamental concepts from abstract algebra and module theory.
  • It involves proving the existence and uniqueness of a maximal submodule with specific regularity properties.
  • Methods include submodule analysis and the investigation of properties under module homomorphisms and direct products.

Main Results:

  • Proved that every unitary R-module A contains a unique maximal GF-regular submodule, denoted as M GF(A).
  • Demonstrated that for any submodule K of A, the maximal GF-regular submodule of K is given by MGF(K) = K ∩ M GF(A).
  • Showed that for a projective R-module A, M GF(A) is a G-pure submodule and MGF(A) = M(R) · A.

Conclusions:

  • The existence and uniqueness of the maximal GF-regular submodule (M GF(A)) are established for unitary R-modules.
  • The interaction between GF-regular submodules and other submodules is clarified through the intersection property.
  • The specific behavior of M GF(A) in projective modules is characterized, linking it to G-purity and the Jacobson radical M(R).