1) Heat oven to ~400*F.
2) Put ring gear in oven for 45-60 minutes (you want to heat it all the way through.)
3) Keep carrier handy.
4) Lower ring gear onto carrier while it's hot.
You'll find it easier to keep aligned if you take 2-3 screws and cut the heads off, then thread them loosely into the holes. Use the screws for alignment pins, then remove (should come out with the fingers.) Install new screws and tighten to spec.
When you heated up the ring gear, you expanded it slightly - probably enough to fit over the carrier. Make sure there's no oil on the ring gear before it goes in the oven!
For a really thorough job of doing this, put the carrier in the freezer (make sure it's dry!) just before or just after the ring gear goes in the oven. Remove the carrier from the freezer just before the ring gear comes out of the oven and goes on - you'll get more room that way (the carrier shrinks in the freezer. It's only a couple of thousandths each way - but it's enough. This is the same way you install flywheel ring gears when they wear out...)
Since condensation is an issue with the carrier coming out of the freezer, turn the oven down to 250* while you're tightening the ring gear screws. Once you've torqued it down (using LocTite #242 or equivalent, by the way. Ring gear screws getting loose really sucks!) put the carrier assembly back in the oven for about ten minutes - the temperature is just enough to boil off any water, but not so much to weaken the LocTite you just put on. It will still cure to full strength.
Remove the carrier, let it cool to room temperature, and proceed with the install. If you don't have a press to get the carrier bearings on, you can put the bearings in the oven with the ring gear - and then handle all the assembly there at once (works for the same reason. You may have to tap the bearings on, and you will have to clean the oil off before you put them in the oven, if you don't want your wife wroth with you...) A little never-seez inside the bearing ID will make installation easier as well. You should be able to get everything done before the temperatures come back together, and lock everything up (once the ring gear is in place, you can put one or two of the new screws in finger tight and focus on the bearings. You don't need to torque the screws while the gear is hot and the carrier cold - just locate the gear properly.)
The typical kitchen oven does not get hot enough to change the temper of pretty much any steel alloy, so you'll be fine. I still consider 400-450* a good "safe" limit, tho (typically, you have to get above 600*F to change the temper of steel alloys.)
Ed - you can eliminate the need for a press to assemble a number of jobs with the careful application and removal of even heat... A temperature difference of some 300-400* makes assembly easier on many jobs!