透视性与透视-施罗德-伯恩斯坦性质
Perspectivity and the perspective-Schröder-Bernstein Property
AI总结：
本文研究模中透视性，证明多类模满足透视-施罗德-伯恩斯坦性质，并给出新刻画及应用。
AI中文摘要：
本文研究了模中透视性的各个方面。我们证明了以下模类满足透视-施罗德-伯恩斯坦性质：具有传递透视性的模、拟连续模、Harada模（因此也是离散模）以及具有（有限）交换性质的拟离散模。此外，我们证明了对于半正则环，施罗德-伯恩斯坦性质蕴含透视-施罗德-伯恩斯坦性质。我们证明了对于$A,B \subseteq ^{\oplus} M$，如果$A$的所有补都与$B$透视，则$B$的所有补都与$A$透视。我们还给出了弱透视模和透视性为传递的模的新刻画。给出了这些结果的一些应用。最后我们证明了在环$R$中透视性是传递的当且仅当$R$的每个特殊清洁元都是透视的。
英文摘要：
In this paper, we study various aspects of perspectivity in modules. We prove that the following classes of modules satisfy the perspective-Schröder-Bernstein property: modules with transitive perspectivity, quasi-continuous modules, Harada (and hence discrete) modules and quasi-discrete modules with the (finite) exchange property. Furthermore, we prove that for a semiregular ring, the Schröder-Bernstein property implies the perspective-Schröder-Bernstein property. We prove that for $A,B \subseteq ^{\oplus} M$, if all complements of $A$ are perspective with $B$, then all complements of $B$ are perspective with $A$. We also provide new characterizations of weakly perspective modules and modules in which perspectivity is transitive. Some applications of these results are given. We finally prove that perspectivity is transitive in a ring $R$ if and only if every special clean element of $R$ is perspective.
