Keeping cat in the house

We are having a problem keeping our cat in the house. Everytime someone opens a door the cat is there ready to dash through it. We try very hard to be aware and prevent it but sometimes it gets through. Once out it stays out all night and we are worried about cars, critters or coyotes getting it. (He was fixed as a kitten.) Does anyone have suggestions on how we can fix this problem. Either by training him or by installing a device or whatever? Thanks...!

Can you build a little vestibule? That's what we did. Our front door opens onto a narrow hallway, and my husband put up a second door a yard or so behind that one. A friend with cats liked it so much that she had one put in, although her front door opens into the living room.

Thanks so much! We do have a screened in porch in the back that he can go out into, but the front door and garage door are the real problems. This is a good suggestion and we'll have to look into it...thanks again.