Chevy check engine light

I have a problem with my Jeep for the last week or so, it won't start. I can shove some starting fluid down it and it will fire, but not run. And when it fires, the check engine light goes off but when it doesn't run, it stays off. Now my question is: isn't that light supposed to come on if the engine dies? I an thinking it should, but can't remember for sure. Just wanting to find out so I can go in the right direction with this. It is not showing any codes. And bear in mind this is a Chevy engine