Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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RINGS AND MODULES WHICH ARE STABLE UNDER NILPOTENTS OF THEIR INJECTIVE HULLS

  • Nguyen Thi Thu Ha (Faculty of Fundamental Science Industrial University of Ho Chi Minh City)
  • Received : 2022.02.15
  • Accepted : 2022.05.04
  • Published : 2023.03.31
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Abstract

It is shown that every nilpotent-invariant module can be decomposed into a direct sum of a quasi-injective module and a square-free module that are relatively injective and orthogonal. This paper is also concerned with rings satisfying every cyclic right R-module is nilpotent-invariant. We prove that R ≅ R1 × R2, where R1, R2 are rings which satisfy R1 is a semi-simple Artinian ring and R2 is square-free as a right R2-module and all idempotents of R2 is central. The paper concludes with a structure theorem for cyclic nilpotent-invariant right R-modules. Such a module is shown to have isomorphic simple modules eR and fR, where e, f are orthogonal primitive idempotents such that eRf ≠ 0.

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Acknowledgement

The author is grateful to the referee for her/his valuable comments. The author would like to thank Prof. Truong Cong Quynh for valuable comments and useful discussion.

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