This paper continues a research program on constructive investigations of non-commutative Ore localizations, initiated in our previous papers, and particularly touches the constructiveness of arithmetics within such localizations. Earlier we have introduced monoidal, geometric and rational types of localizations of domains as objects of our studies. Here we extend this classification to rings with zero divisors and consider Ore sets of the mentioned types which are commutative enough: such a set either belongs to a commutative algebra or it is central or its elements commute pairwise. By using the systematic approach we have developed before, we prove that arithmetic within the localization of a commutative polynomial algebra is constructive and give the necessary algorithms. We also address the important question of computing the local closure of ideals which is also known as the desingularization, and present an algorithm for the computation of the symbolic power of a given ideal in a commutative ring. We also provide algorithms to compute local closures for certain non-commutative rings with respect to Ore sets with enough commutativity.

本文继续了我们先前论文中发起的关于非可交换矿石局部化的建设性研究的研究计划，特别是涉及这种局部化中算术的建构性。较早之前，我们将域的局部化，几何和有理类型引入作为研究对象。在这里，我们将分类扩展到零除数为零的环，并考虑上述类型的Ore集，这些Ore集具有可交换性：此类集合属于可交换代数，或者位于中心，或者其元素成对交换。通过使用我们之前开发的系统方法，我们证明了交换多项式代数的局部化内的算术是建设性的，并给出了必要的算法。我们还解决了计算理想的局部闭合的重要问题，也称为去奇点化，并提出了一种算法，用于计算交换环中给定理想的符号幂。我们还提供了一些算法，可针对具有足够可交换性的矿石集计算某些非可交换环的局部闭合。