In this manuscript, we transport the classical Operator Theory on complex Banach spaces to normed modules over absolutely valued rings. In some cases, we are able to extend classical results on complex Banach spaces to normed modules over normed rings. In order to make sure that bounded linear maps on normed modules coincide with the continuous linear maps, it is sufficient that the underlying ring be practical (a topological ring is practical if the invertibles approach zero). In order for other classical results to work on the scope of normed modules, it is sufficient that the underlying ring be normalizing. Normalizing rings is a new class of rings introduced in this manuscript. We provide a characterization of such rings as well as nontrivial examples.

中文翻译：

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归一化环

在本手稿中，我们将关于复杂Banach空间的经典算子理论传输到绝对值环上的赋范数。在某些情况下，我们能够将复杂Banach空间上的经典结果扩展到范环上的范模块。为了确保范数模块上的有界线性图与连续线性图重合，底层环是可行的（如果可逆数接近零，则拓扑环是可行的）就足够了。为了使其他经典结果在范数模块的范围内起作用，使基础环规范化就足够了。归一化环是本手稿中引入的一类新的环。我们提供了此类环的表征以及非平凡的示例。