Best answer: Amazon and Ring products can work together through Alexa, but you can't directly integrate your Amazon Cloud Cam into your Ring system nor have them talk directly to each other. Amazon Cloud Cam is its own separate security system.
Two systems, one master
Amazon bought Ring
Plan Price Device limit Custom zones Days recorded
Cloud Cam Free Free 3 cams 1 day
Cloud Cam Basic $7/mo
$70/yr 3 cams 7 days
Cloud Cam Extended $10/mo
$100/yr 5 cams 14 days
Cloud Cam Pro $20/mo
$200/yr 10 cams 30 days
Ring Free Free —
Ring Protect Basic $3/mo
$30/yr 1 cam 60 days
Ring Protect Plus $10/mo
$100/yr All @ single address 60 days
Ring is a much larger and more comprehensive home security system, with a wide array of cameras, sensors, and even alarm systems, but it also has more expensive cameras and a subscription is required for Ring cameras to record anything. Ring's systems are completely independent of the Amazon Cloud Cam system, so if you were hoping to save a few bucks with an Amazon Cloud Cam over the new Ring Stick Up Cam, sorry to say that it won't integrate with the rest of the Ring system the way you want it to.
Alexa in the middle
Amazon Alexa integration is a highly prized feature of both the Ring and Amazon Cloud Cam systems, and through Amazon Alexa and its smart home configurations, you can use Ring and Amazon Cloud Cam together in one home with Alexa serving as a central hub for it all. The Ring system and the Amazon Cloud Cam systems can't talk to each other, but they can each talk to Alexa or directly to you.
If one of your Ring sensors detects something, you can ask Alexa to put up the feed from a nearby Cloud Cam with the "Show me [camera name]" command. The motion alert going off on a Ring camera can't tell the Amazon camera to record directly, nor can an Amazon Cloud Cam motion detection trigger the sirens on a Ring alarm system. Both systems can work for you, even if they can't technically work together.
Amazon Cloud Cam
It only integrates with Alexa, but that might be enough
Amazon's camera can't play directly with Ring, but its Alexa integration can allow you to use both systems in concert. If you're looking for a basic security camera that won't break your budget and don't have a huge space to cover, the Cloud Cam might be a perfect fit for your dorm room or apartment.