In this investigation we give a module-theoretic counterpart of the well known De Morgan’s laws for rings and topological spaces. We observe that the module-theoretic De Morgan’s laws are related with semiprime modules and modules in which the annihilator of any fully invariant submodule is a direct summand. Also, we give a general treatment of De Morgan’s laws for ordered structures (idiomatic-quantales). At the end, the manuscript goes back to the ring theoretic realm, in this case we study the non-commutative counterpart of Dedekind domains, and we describe Asano prime rings using the strong De Morgan law.

中文翻译：

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关于模块的德摩根定律

在这项研究中，我们给出了著名的 De Morgan 环和拓扑空间定律的模理论对应物。我们观察到模理论德摩根定律与半素模和模相关，其中任何完全不变子模的湮灭器都是直接被加数。此外，我们对有序结构（惯用量子数）的德摩根定律进行了一般处理。最后，手稿回到了环理论领域，在这种情况下，我们研究了 Dedekind 域的非交换对应物，并使用强德摩根定律描述了 Asano 素环。