AI 中文总结
研究具有整数系数的列有限矩阵环M，证明M-模构成特定范畴，引入M-环概念，表明M-模（环）范畴等价于Clausen和Scholze的轻实阿贝尔群（环）范畴并提供新视角。
AI 中文摘要
我们考虑具有整数系数的列有限矩阵环M。我们证明M-模构成一个加法封闭的对称幺半阿贝尔范畴，它本质上包含完全可度量化的线性拓扑阿贝尔群作为一个满子范畴。我们还引入并讨论了M-环的概念。最后，我们表明M-模（分别为M-环）的范畴实际上等价于Clausen和Scholze的轻实阿贝尔群（分别为轻实环）的范畴，这对专家来说不足为奇。然而，我们认为我们更直接的方法可能为他们的理论提供新视角。
英文摘要
We consider the ring M of column-finite matrices with integer coefficients. We prove that M-modules form an additive closed symmetric monoidal abelian category that essentially contains complete metrizable linearly topologized abelian groups as a full subcategory. We also introduce and discuss the notion of an M-ring. In the end, we show that the category of M-modules (resp. M-rings) is actually equivalent to the category of light solid abelian groups (resp. light solid rings) of Clausen and Scholze, which should come as no surprise to specialists. However, we believe that our more straightforward approach may offer a fresh perspective on their theory.
