  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.
