I agree with the above, I personally would not terminate that rye. Do as dipper and dogghr suggest and leave it go to maturity. Mowing it alone will not kill it unless you let it start to produce seed heads, it is a grass and will keep regrowing like your lawn for most of the summer, until it would be at the age of "maturity" which would be sometime in July most likely, and then it would start to brown down and die. If that is what you are going to do anyway, you might as well let it grow and become fawning and brooding cover for the deer and turkeys and roll it or mow it to get a nice mulch and free cover crop for your fall planting. Broadcast(or drill) your fall seed into the standing rye after it has gone to seed and then just mow it short or roll it down with a packer and watch the magic! The fall clover, oats, brassicas, AWP, or whatever else you throw in there will love the moisture holding cover the rye straw will provide and will have no problem pushing right through it once it germinates, it will grow right along side the new free rye seed that you "planted" with your mower/roller .