대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제60권4호
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  • Pages.1071-1083
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  • 2023
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  • 1015-8634(pISSN)
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  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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ON SEMI-REGULAR INJECTIVE MODULES AND STRONG DEDEKIND RINGS

  • Renchun Qu (Department of Basic Courses Chengdu Aeronautic Polytechnic)
  • 투고 : 2022.07.26
  • 심사 : 2022.10.28
  • 발행 : 2023.07.31
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초록

The main motivation of this paper is to introduce and study the notions of strong Dedekind rings and semi-regular injective modules. Specifically, a ring R is called strong Dedekind if every semi-regular ideal is Q0-invertible, and an R-module E is called a semi-regular injective module provided Ext1R(T, E) = 0 for every 𝓠-torsion module T. In this paper, we first characterize rings over which all semi-regular injective modules are injective, and then study the semi-regular injective envelopes of R-modules. Moreover, we introduce and study the semi-regular global dimensions sr-gl.dim(R) of commutative rings R. Finally, we obtain that a ring R is a DQ-ring if and only if sr-gl.dim(R) = 0, and a ring R is a strong Dedekind ring if and only if sr-gl.dim(R) ≤ 1, if and only if any semi-regular ideal is projective. Besides, we show that the semi-regular dimensions of strong Dedekind rings are at most one.

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참고문헌

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