you can also just cut it very low to the soil surface and it will die back, a mower that you can set low is great as it will mulch it a bit, but you can also use a scythe or weed wacker. these will leave longish stems (depending how long the rye is) that will be like straw mulch if you don't till them in. if you till them in, or even stab them in with a shovel, they will digest into the soil over time. covering with a tarp will speed up digestion process and also assure rye is completely dead.
