论文标题
表征$ s $ -Artinianness通过统一性
Characterizing $S$-Artinianness by uniformity
论文作者
论文摘要
令$ r $为具有身份的可交换戒指，而$ s $是$ r $的乘法子集。如果有$ r $ -Module $ m $ $ $ $ $ s $ $ s $ - $ - $ u $ - $ s $ -Artinian用于缩写）模块，如果S $中有$ s \，以至于任何下降的$ m $ $ $ $ $ s $ s $ s $ s $ sub-s $ as $ s $。 $ u $ - $ s $ -Artinian模块的特征是（$ s $ -min） - 条件和$ u $ - $ s $ -s $ -cofinite Properties。如果$ r $本身是$ u $ - $ s $ -S $ -ARTINIAN模块，我们称为$ u $ r $是$ u $ - $ s $ -Artinian戒指，然后证明任何$ u $ - $ s $ s $ s $ semisimple ring is $ u $ u $ - $ s $ s $ - 阿蒂尼亚人。事实证明，$ r $是$ u $ - $ s $ -Artinian，仅当$ r $是$ r $是$ u $ - $ u $ - $ s $ -noetherian，$ u $ - $ u $ - $ s $ -s $ - jacobson gadical $ {\ rm jac} _s jac} _s _s（r）$ r $ r $是$ s $ s $ s $ - nillpotent and $ rm/jac a jac（rm jac a jac（rm） $ u $ - $ s/{\ rm jac} _s（r）$ - 半圣戒指。此外，还提供了一些示例来区分Artinian戒指，$ U $ - $ S $ -ARTINIAN RING和$ S $ -ARTINIAN RINGS。
Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. An $R$-module $M$ is said to be a uniformly $S$-Artinian ($u$-$S$-Artinian for abbreviation) module if there is $s\in S$ such that any descending chain of submodules of $M$ is $S$-stationary with respect to $s$. $u$-$S$-Artinian modules are characterized in terms of ($S$-MIN)-conditions and $u$-$S$-cofinite properties. We call a ring $R$ is a $u$-$S$-Artinian ring if $R$ itself is a $u$-$S$-Artinian module, and then show that any $u$-$S$-semisimple ring is $u$-$S$-Artinian. It is proved that a ring $R$ is $u$-$S$-Artinian if and only if $R$ is $u$-$S$-Noetherian, the $u$-$S$-Jacobson radical ${\rm Jac}_S(R)$ of $R$ is $S$-nilpotent and $R/{\rm Jac}_S(R)$ is a $u$-$S/{\rm Jac}_S(R)$-semisimple ring. Besides, some examples are given to distinguish Artinian rings, $u$-$S$-Artinian rings and $S$-Artinian rings.
