论文标题
$σ$ -semi-compact环和模块
$Σ$-semi-compact rings and modules
论文作者
论文摘要
在本文中，给出了半混合模块的几种特征。除其他结果外，我们研究了半压缩模块具有含义的环。我们介绍了用于模块的属性$σ$ -semi-compact，并表征满足此属性的模块。特别是，我们表明，当$ r $满足左侧（右（分别）右）环上的上升（分别降序）链条条件时，只有$σ$ -semi-compact留下$σ$ -semi-compact。此外，我们证明，当且仅当$ r $留下$σ$ -Semi-compact时，每个纯$ r $ module都是半压缩的。我们还表明，只有在每个纯射击都留下$ r $ - 模块是半紧凑的时，就会留下$ r $。最后，我们考虑了其平坦模块有限（单独）投影的戒指。对于任何带有商环$ Q $的交换性算术圈$ r $，我们证明，当每个flat $ r $ module都是半压缩时，并且仅当每个flat $ r $ r $ z模块都是有限（单独的）投影时，并且只有$ q $是纯半imimimple的。对于Space $ \ Mathrm {min} \ r $ compact，可为减少的交换环$ r $获得类似的结果。我们还证明，如果$ r $剩下$σ$ -semi-compact，则每个$（\ aleph_ {0}，1）$ - flat flat flat左$ r $ -module是单一的投影，并且如果$ r^{\ r^{\ mathbb {n}} $ is $ r^{\ r^{\ r^{\ r^{
In this paper several characterizations of semi-compact modules are given. Among other results, we study rings whose semi-compact modules are injective. We introduce the property $Σ$-semi-compact for modules and we characterize the modules satisfying this property. In particular, we show that a ring $R$ is left $Σ$-semi-compact if and only if $R$ satisfies the ascending (resp. descending) chain condition on the left (resp. right) annulets. Moreover, we prove that every flat left $R$-module is semi-compact if and only if $R$ is left $Σ$-semi-compact. We also show that a ring $R$ is left Noetherian if and only if every pure projective left $R$-module is semi-compact. Finally, we consider rings whose flat modules are finitely (singly) projective. For any commutative arithmetical ring $R$ with quotient ring $Q$, we prove that every flat $R$-module is semi-compact if and only if every flat $R$-module is finitely (singly) projective if and only if $Q$ is pure semisimple. A similar result is obtained for reduced commutative rings $R$ with the space $\mathrm{Min}\ R$ compact. We also prove that every $(\aleph_{0},1)$-flat left $R$-module is singly projective if $R$ is left $Σ$-semi-compact, and the converse holds if $R^{\mathbb{N}}$ is an $(\aleph_{0},1)$-flat left $R$-module.
