ON SEMI-REGULAR INJECTIVE MODULES AND STRONG DEDEKIND RINGS
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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 $\Ext^1_R(T,E)=0$ 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$(R)\leq 1$, 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.
Abstract
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 $\Ext^1_R(T,E)=0$ 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$(R)\leq 1$, 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.
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