ANNUAL, not perennial, rye grasses are sometimes planted to give lawns like zoysia and bermuda---that turn brown during the dormant season---some green during the period. The good news is the seed is dirt cheap BUT make sure you get quality seed with low weed content stated for it. A nursery or turf seed company will have it. The contractor stuff some use to establish some sort of lawns quickly typically has lots of seeds you do not want.
To achieve any degree of success you are going to have to dethatch the turf and/or power rake it, broadcast the seed, top dress and water frequently until the seeds germinate. Annual rye germinates quickly but will be slowed by cooler temperatures. If this is new sod, you will do more damage than good dethatching and power raking.
Some golf courses with bermuda grass dye the dormant turf. To me it looks like someone put green food coloring on it all.
I had a massive bermuda front lawn in Northern California that looked spectacular during the growing season and dead when dormant. I never had much luck overseeding with rye but if you are game, give it a shot. Like it say, it will not set you back much money to try one season.
If it takes, you will want to mow the rye taller than zoysia or bermuda until it comes out of dormancy.