symmetric monoidal (∞,1)-category of spectra
Given a (possibly noncommutative) unital ring there are many situations when certain elements or matrices can be inverted in a universal way obtaining a new “localized” ring equipped with a localization homomorphism under which all elements in are mapped to multiplicatively invertible elements (units). The latter property must be modified for Cohn localization at multiplicative set of matrices.
We can typically invert elements in a left or right Ore subset or much more generally some multiplicative set or matrices (Cohn localization) etc. There are also some specific localizations like Martindale localizations in ring theory.