论文标题
关于乘法FS模型和尺寸对称性
On multiplication fs-modules and dimension symmetry
论文作者
论文摘要
在本文中，我们首先研究了$ fs $ - 模型，即具有有限的小型子模块。 我们表明，具有有限空心尺寸的每个$ fs $ - 模块都是notherian。 另外，我们证明$ r $ -module $ m $带有有限的黄金尺寸，当$ fs $ -module时，并且仅当$ m = m_1 \ oplus m_2 $，其中$ m_1 $ as s emimisimple和$ m_2 $ as $ m_2 $ is $ fs $ fs $ -module at $ soc（m_2）$ soc（m_2）\ ll m $ $ $ $。然后，我们调查乘法$ fs $模型对交换戒指，并在且仅当每个乘法$ r $ -module是$ fs $ -module时，就表明$ r $是$ fs $ ring。特别是，我们证明$ r $ -submodules $ m $和$ s $ s $ submodules的晶格是$ m $的，其中$ s = end_r（m）$。因此，$ m_r $和$ _sm $具有Krull（Noetherian，Goldie和Hollow）的相同尺寸。此外，我们证明，对于任何自发乘积模块$ m $，要为$ fs $ - 模块作为正确的$ r $ - 模块，并且作为左$ s $ module等效。
In this paper, we first study $fs$-modules, i.e., modules with finitely many small submodules. We show that every $fs$-module with finite hollow dimension is Noetherian. Also, we prove that an $R$-module $M$ with finite Goldie dimension, is an $fs$-module if and only if $M = M_1 \oplus M_2$, where $M_1$ is semisimple and $M_2$ is an $fs$-module with $Soc(M_2) \ll M$. Then, we investigate multiplication $fs$-modules over commutative rings and show that $R$ is an $fs$-ring if and only if every multiplication $R$-module is an $fs$-module. In particular, we prove that the lattices of $R$-submodules of $M$ and $S$-submodules of $M$ are coincide, where $S=End_R(M)$. Consequently, $M_R$ and $_SM$ have the same dimension of Krull (Noetherian, Goldie and hollow). Further, we prove that for any self-generator multiplication module $M$, to be an $fs$-module as a right $R$-module and as a left $S$-module are equivalent.
