$\scr{Z^{ \ast}}$ - semilocal modules and the proper class $\scr{RS}$
Over an arbitrary ring, a module $M$ is said to be $\scr{Z^{ \ast}}$ -semilocal if every submodule $U$ of $M$ has a $\scr{Z^{ \ast}}$ -supplement $V$ in $M$, i.e., $M = U + V$ and $U \cap V \subseteq \scr{Z^{ \ast}} (V )$, where $\scr{Z^{ \ast}}(V ) = \{m \in V | Rm$ is a small module $\}$ is...
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| Date: | 2019 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2019
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1447 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| Summary: | Over an arbitrary ring, a module $M$ is said to be $\scr{Z^{ \ast}}$ -semilocal if every submodule $U$ of $M$ has a $\scr{Z^{ \ast}}$ -supplement $V$
in $M$, i.e., $M = U + V$ and $U \cap V \subseteq \scr{Z^{ \ast}} (V )$, where $\scr{Z^{ \ast}}(V ) = \{m \in V | Rm$ is a small module $\}$ is the $\mathrm{R}\mathrm{a}\mathrm{d}$-small
submodule. In this paper, we study basic properties of these modules as a proper generalization of semilocal modules. In particular, we show that the class of $\scr{Z^{ \ast}}$ -semilocal modules is closed under submodules, direct sums, and factor modules.
Moreover, we prove that a ring $R$ is $\scr{Z^{ \ast}}$ -semilocal if and only if every injective left R-module is semilocal. In addition,
we show that the class $\scr{RS}$ of all short exact sequences $E :0 \xrightarrow{\psi} M \xrightarrow{\phi} K \rightarrow 0$ such that $\mathrm{I}\mathrm{m}(\psi )$ has a
$\scr{Z^{ \ast}}$ -supplement in $N$ is a proper class over left hereditary rings. We also study some homological objects of the proper
class $\scr{RS}$ . |
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