Bull. Korean Math. Soc. 2023; 60(4): 1071-1083
Online first article July 4, 2023 Printed July 31, 2023
https://doi.org/10.4134/BKMS.b220516
Copyright © The Korean Mathematical Society.
Renchun Qu
Chengdu Aeronautic Polytechnic
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 $Q_0$-invertible, and an $R$-module $E$ is called a semi-regular injective module provided ${\rm Ext}^1_R(T,E)=0$ for every $\mathcal{Q}$-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 ${\rm DQ}$-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$(R)\leq 1$, 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.
Keywords: Strong Dedekind ring, $\DQ$-ring, semi-regular injective module, semi-regular global dimension
MSC numbers: 13F05, 13C11
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