RINGS AND MODULES WHICH ARE STABLE UNDER NILPOTENTS OF THEIR INJECTIVE HULLS
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
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 $R\cong R_1\times R_2$, where $R_1, R_2$ are rings which satisfy $R_1$ is a semi-simple Artinian ring and $R_2$ is square-free as a right $R_2$-module and all idempotents of $R_2$ 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 $eRf\ne 0$.
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 $R\cong R_1\times R_2$, where $R_1, R_2$ are rings which satisfy $R_1$ is a semi-simple Artinian ring and $R_2$ is square-free as a right $R_2$-module and all idempotents of $R_2$ 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 $eRf\ne 0$.
참고문헌 (19)
- A. Abyzov, "Dual automorphism-invariant modules over perfect rings", Sib. Math. J, 58, pp.743~751 (2017)
- N. V. Dung, "Pitman Research Notes in Math", pp.313~ (1994), Longman
- Noyan Er, "Rings and modules which are stable under automorphisms of their injective hulls", Journal of Algebra, 379, pp.223~229 (2013), Elsevier BV[DOI]
- A. Facchini, "Module Theory", pp.1998, Birkhauser
- Pedro A. Guil Asensio, "Additive unit structure of endomorphism rings and invariance of modules", Bulletin of Mathematical Sciences, 7(2), pp.229~246 (2016), World Scientific Pub Co Pte Ltd[DOI]
- DINESH KHURANA, "RIGHT SELF-INJECTIVE RINGS IN WHICH EVERY ELEMENT IS A SUM OF TWO UNITS", Journal of Algebra and Its Applications, 06(02), pp.281~286 (2007), World Scientific Pub Co Pte Lt[DOI]
- Muhammet Tamer Kosan, "Independently generated modules", 대한수학회보, 46(5), pp.867~871 (2009), 대한수학회
- M. Tamer Koşan, "Nilpotent-invariant modules and rings", Communications in Algebra, 45(7), pp.2775~2782 (2016), Informa UK Limited[DOI]
- M. Tamer Koşan, "Rings whose (proper) cyclic modules have cyclic automorphism-invariant hulls", Applicable Algebra in Engineering, Communication and Computing, 32(3), pp.385~397 (2021), Springer Science and Business Media LLC[DOI]
- M. Tamer Koşan, "Kernels of homomorphisms between uniform quasi-injective modules", Journal of Algebra and Its Applications, 21(08) (2021), World Scientific Pub Co Pte Ltd[DOI]
- M. Tamer Koşan, "Rings with each right ideal automorphism-invariant", Journal of Pure and Applied Algebra, 220(4), pp.1525~1537 (2016), Elsevier BV[DOI]
- TSIU-KWEN LEE, "MODULES WHICH ARE INVARIANT UNDER AUTOMORPHISMS OF THEIR INJECTIVE HULLS", Journal of Algebra and Its Applications, 12(02), pp.1250159~ (2012), World Scientific Pub Co Pte Lt[DOI]
- S. H. Mohamed, "London Mathematical Society Lecture Note Series", pp.147~ (1990), Cambridge University Press
- Pace P. Nielsen, "Square-free modules with the exchange property", Journal of Algebra, 323(7), pp.1993~2001 (2010), Elsevier BV[DOI]
- Truong Cong Quynh, "Rings Characterized via Some Classes of Almost-Injective Modules", Bulletin of the Iranian Mathematical Society, 47(5), pp.1571~1584 (2020), Springer Science and Business Media LLC[DOI]
- Truong Cong Quynh, "Modules close to the automorphism-invariant and coinvariant", Journal of Algebra and Its Applications, 18(12), pp.1950235~ (2019), World Scientific Pub Co Pte Lt[DOI]
- Truong Cong Quynh, "Modules which are invariant under nilpotents of their envelopes and covers", Journal of Algebra and Its Applications, 20(12) (2020), World Scientific Pub Co Pte Ltd[DOI]
- Truong Cong Quynh, "Rings all of whose finitely generated ideals are automorphism-invariant", Journal of Algebra and Its Applications, 21(08) (2021), World Scientific Pub Co Pte Ltd[DOI]
- Truong Cong Quynh, "On automorphism-invariant modules", Journal of Algebra and Its Applications, 14(05), pp.1550074~ (2015), World Scientific Pub Co Pte Lt[DOI]