Localising modules, rings, and spectra.

Given a ring R and a multiplicative subset S of R, then one can localise R at S using a number of different constructions. Using some of these constructions it is not clear the result is flat over R, or non-zero, or even a ring. Commutative algebra tells us there is nothing to worry about in this situation, but as soon as we step into non-commutative algebra we require some type of Ore condition. 

We are going to take a step in another direction, towards equivariant homotopical algebra, and see that in the world of equivariant ring spectra, there is a necessary and sufficient condition to localise a ring sensibly. This follows work by Hill and Hopkins, and can sometimes be phrased in the global homotopy theory language of Schwede.