Abstract The Cohn localization is a process of adjoining universal inverse morphisms to the category of R-modules over a noncommutative ring R. Finding specific models for universal localizations is, in general, difficult. For certain classes of rings, however, there exist constructions that are more accessible. We define such a class of rings, the generalized triangular matrix rings, and provide complete descriptions of their universal localizations with respect to certain classes of morphisms, generalizing results of Schofield and Sheiham. These descriptions are themselves generalized matrix rings. The results demonstrate, in particular, that localizing triangular matrix rings produces matrix rings with an increased degree of symmetry. This explains, in part, why Cohn localizations of triangular 2 × 2 matrix rings are always full matrix rings. However, the universal localization of a triangular matrix ring of order greater than two is often not fully symmetric and thereby not a full matrix ring. One consequence of this incomplete symmetry is the lack of a full set of matrix units to identify the localization. In spite of this difficulty, the localization does contain enough idempotents so that its entry bimodules can be recognized. In the case where the set of morphisms forms a maximal tree, the universal localization of any triangular matrix ring is a full matrix ring.

中文翻译：

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某些矩阵环的通用定位

摘要 Cohn 定位是在非交换环 R 上将通用逆态射与 R 模的范畴相连接的过程。寻找通用定位的特定模型通常很困难。然而，对于某些类别的环，存在更易于访问的结构。我们定义了这样一类环，即广义三角矩阵环，并提供了它们关于某些类态射的通用定位的完整描述，概括了 Schofield 和 Sheiham 的结果。这些描述本身就是广义的矩阵环。结果特别表明，定位三角形矩阵环会产生对称度增加的矩阵环。这部分解释了为什么三角形 2 × 2 矩阵环的 Cohn 局部化总是完整的矩阵环。然而，大于二阶的三角形矩阵环的全域定位通常不是完全对称的，因此不是一个完整的矩阵环。这种不完全对称的一个后果是缺乏一套完整的矩阵单元来识别定位。尽管有这个困难，但本地化确实包含足够的幂等项，因此可以识别其入口双模块。在态射集合形成极大树的情况下，任何三角形矩阵环的全域定位都是一个完整的矩阵环。本地化确实包含足够的幂等项，因此可以识别其入口双模块。在态射集合形成极大树的情况下，任何三角形矩阵环的全域定位都是一个完整的矩阵环。本地化确实包含足够的幂等项，因此可以识别其入口双模块。在态射集合形成极大树的情况下，任何三角形矩阵环的全域定位都是一个完整的矩阵环。