论文标题
拓扑半精美的拓扑环
Topologically semiperfect topological rings
论文作者
论文摘要
我们定义了拓扑半精感（完整，分离，右线性）拓扑环，并通过等效条件来表征它们。我们表明，当且仅当模块可作为具有局部内态环的模块的（无限）直接直接总和时，具有有限拓扑的模块的内态环在拓扑上是半截面的。然后，我们研究了拓扑半精美的拓扑环的结构特性，并证明其拓扑雅各布森自由基是强烈封闭的，相关的拓扑用环在拓扑上是半学的。对于具有局部内态环的直接模块的内态环，拓扑雅各布森激进分子被明确描述为所有非同构矩阵的集合。此外，我们证明，在拓扑半专用的拓扑环上，所有有限生成的离散模块都具有模块类别的投影覆盖，而所有晶格 - 芬特式的禁用模块都在两个模块类别和游乐类别的类别中都具有投影覆盖。我们还表明，拓扑上半部拓扑环的拓扑雅各布森（Jacobson）激进等于雅各布森（Jacobson）激进分子的闭合，并提出一个反例，表明拓扑雅各布森（Jacobson）激进的激进性可能比抽象大。最后，我们讨论了提升dempotents的问题，使拓扑结构雅各布森激进分子和拓扑上半部拓扑环的射击构造结构。
We define topologically semiperfect (complete, separated, right linear) topological rings and characterize them by equivalent conditions. We show that the endomorphism ring of a module, endowed with the finite topology, is topologically semiperfect if and only if the module is decomposable as an (infinite) direct sum of modules with local endomorphism rings. Then we study structural properties of topologically semiperfect topological rings and prove that their topological Jacobson radicals are strongly closed and the related topological quotient rings are topologically semisimple. For the endomorphism ring of a direct sum of modules with local endomorphism rings, the topological Jacobson radical is described explicitly as the set of all matrices of nonisomorphisms. Furthermore, we prove that, over a topologically semiperfect topological ring, all finitely generated discrete modules have projective covers in the category of modules, while all lattice-finite contramodules have projective covers in both the categories of modules and contramodules. We also show that the topological Jacobson radical of a topologically semiperfect topological ring is equal to the closure of the abstract Jacobson radical, and present a counterexample demonstrating that the topological Jacobson radical can be strictly larger than the abstract one. Finally, we discuss the problem of lifting idempotents modulo the topological Jacobson radical and the structure of projective contramodules for topologically semiperfect topological rings.
