Kesa gatame - in depth instruction (via BeyondGrappling)
Above is a gif of Ikuhisa Minowa using the Kesa Gatame to attack an Ude Garami against Baru Harn at Dream 17. You can see Minowa force Harn's right arm down to be entangled in his legs. Minowa hooks his left knee over Harn's wrist and then crosses his legs, kicking them back. At the same time, Minowa pulls up on Harn's head, putting immense pressure on Harn's shoulder. If Harn does not tap, his shoulder will begin to tear.
In the first thirty seconds or so of her first UFC title match, Rousey came very close to pulling this off against Carmouche.
After a takedown, Rousey lands in Kesa Gatame and kicks her right leg over Carmouche's wrist, setting up the Ude Garami. The problem, however is that they are against the cage and Carmouche is on her side. She's not fully pinned in the Kesa Gatame. This allows her to pull her arm of out of the submission and leaves Ronda with just control of the head.
At that point, Ronda is left with a simple head lock. Once Carmouche is able to slip her head free, the path to Ronda's back is wide open. As you can see here, Carmouche rolls up, frees her arm, and is able to take Rousey's back.
Fans talk quite a bit of how Ronda gives up her back, but it has only happened twice in her MMA career: once in a scramble off a sweep of Meisha Tate and this time from Kesa Gatame. The problem with the traditional Kesa Gatame is the grip on the near side arm. It is vital to protect the back. In judo, you can grip the gi to help control the arm, but not in MMA.
If Ronda is concerned about Tate getting on her back, look for her to make greater use of the Kuzure Kesa Gatame where, instead of grabbing the head, Rousey would underhook the far arm preventing any back take. The position has become very popular in sport jiu jitsu. It provides good control and ability to transition between side control positions and still allows for submission attacks. It doesn't put down the same amount of horrific pressure and doesn't fully take away the bridge.
Judo techniques Kuzure Kesa Gatame (via Judo techniques)
It is an excellent arrow in the quiver for Rousey and as it has been ingrained into her since her childhood, it has served her very well in MMA thus far. It also gave Meisha Tate fits in their first fight.
For more on Ronda Rousey's armbar check out Ben Thapa's Judo Chops:
Also check out the BJJ Scout's excellent video on her use of inside trips