As the steel ring cools, it thermally contracts. The steel ring will be in tension and the pipe on to which it shrinks will be in compression.
The key about thermal strain is that it does produces displacement without stress.
One needs the IDs/ODs of the pipe and steel ring, and the starting and final temperatures of the steel ring. One can simply the problem by assuming the pipe does not change temperature from its initial temperature (e.g. room temp) and this is the final temperature of the steel ring.
When two dissimilar metals are mated, one must be concerned with differential thermal expansion - a consequence of different thermal expansion coefficients and/or thermal gradients.