通过环的加法子群生成理想
Generating ideals by additive subgroups of rings
AI总结：
该研究通过环的加法子群生成理想，证明了有限指数理想和加法子群的生成性质，并回答了关于博斯紧致化的相关问题。
摘要：
我们获得了关于有限指数理想和环的加法子群以及环的模型论连通分量的几个基本结果，这些结果涉及在环的加法群中通过有限步骤生成。令R为任意配备任意附加一阶结构的环，A为一个参数集。我们证明，当H是(R,+)中的A-可定义、有限指数子群时，H+RH包含一个A-可定义、双边有限指数理想。作为推论，我们回答了[群和环的博斯紧致化，J. Gismatullin, G. Jagiella和K. Krupiński]中的问题3.9：如果R是单环，则(\ar R,+)^{00}_A + \ar R·(\ar R,+)^{00}_A + \ar R·(\ar R,+)^{00}_A = \ar R^{00}_A，其中\ar R≻R是R的足够饱和的初等扩张，且(\ar R,+)^{00}_A[resp. \ar R^{00}_A]是\ar R中最小的A-型可定义、有限指数加法子群[resp.理想]。这表明\ar R^{00}_A=\ar R^{000}_A，其中\ar R^{000}_A是\ar R中最小的A-不变、有限指数理想。如果R具有有限特征（不一定是单环），我们得到更精确的结果：(\ar R,+)^{00}_A + \ar R·(\ar R,+)^{00}_A = \ar R^{00}_A。我们还获得了关于有限生成（不一定是单环）环和拓扑环的类似结果。上述结果表明，之前论文中获得的三角群上单环的简化博斯紧致化描述（见上述论文的推论3.5）对所有单环都有效。我们分析了许多例子，其中我们计算了通过(\ar R ∪ {1})·(\ar R,+)^{00}_A生成群所需的步骤数，并研究了相关方面，展示了我们主要结果的“最优性”并回答了一些自然问题。
英文摘要：
We obtain several fundamental results on finite index ideals and additive subgroups of rings as well as on model-theoretic connected components of rings, which concern generating in finitely many steps inside additive groups of rings. Let $R$ be any ring equipped with an arbitrary additional first order structure, and $A$ a set of parameters. We show that whenever $H$ is an $A$-definable, finite index subgroup of $(R,+)$, then $H+RH$ contains an $A$-definable, two-sided ideal of finite index. As a corollary, we positively answer Question 3.9 of [Bohr compactifications of groups and rings, J. Gismatullin, G. Jagiella and K. Krupiński]: if $R$ is unital, then $(\bar R,+)^{00}_A + \bar R \cdot (\bar R,+)^{00}_A + \bar R \cdot (\bar R,+)^{00}_A = \bar R^{00}_A$, where $\bar R \succ R$ is a sufficiently saturated elementary extension of $R$, and $(\bar R,+)^{00}_A$ [resp. $\bar R^{00}_A$] is the smallest $A$-type-definable, bounded index additive subgroup [resp. ideal] of $\bar R$. This implies that $\bar R^{00}_A=\bar R^{000}_A$, where $\bar R^{000}_A$ is the smallest invariant over $A$, bounded index ideal of $\bar R$. If $R$ is of finite characteristic (not necessarily unital), we get a sharper result: $(\bar R,+)^{00}_A + \bar R \cdot (\bar R,+)^{00}_A = \bar R^{00}_A$. We obtain similar results for finitely generated (not necessarily unital) rings and for topological rings. The above results imply that the simplified descriptions of the definable (so also classical) Bohr compactifications of triangular groups over unital rings obtained in Corollary 3.5 of the aforementioned paper are valid for all unital rings. We analyze many examples, where we compute the number of steps needed to generate a group by $(\bar R \cup \{1\}) \cdot (\bar R,+)^{00}_A$ and study related aspects, showing "optimality" of some of our main results and answering some natural questions.
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