论文标题
无限和有限直接总和模块的基本结果
An elementary result on infinite and finite direct sums of modules
论文作者
论文摘要
令$ r $为戒指，并考虑一个用两个（通常是无限）直接分解给出的左$ r $ - 模块，$ a \ oplus（\ bigoplus_ {i} in i} c_i）= m = m = b \ oplus（\ bigoplus_有限生成。我们表明，存在有限的子集$ i_0 \ subseteq i，$ $ j_0 \ subseteq j，$和直接的summand $ y \ y \ subseteq \ bigoplus_ {i \ in I_0} c_i，$ 然后，我们注意到这种结果可以并且无法概括的一些方法，以及一些相关的问题。
Let $R$ be a ring, and consider a left $R$-module given with two (generally infinite) direct sum decompositions, $A\oplus(\bigoplus_{i\in I} C_i)=M=B\oplus(\bigoplus_{j\in J} D_j),$ such that the submodules $A$ and $B$ and the $D_j$ are each finitely generated. We show that there then exist finite subsets $I_0\subseteq I,$ $J_0\subseteq J,$ and a direct summand $Y\subseteq \bigoplus_{i\in I_0} C_i,$ such that $A \oplus Y \ =\ B \oplus(\bigoplus_{j\in J_0} D_j).$ We then note some ways that this result can and cannot be generalized, and some related questions.
