Whether or not it works, trying to kill an already large tree by ring-barking, poison or whatever, can create an even bigger hazard if it later falls on something or someone, or a potentially costlier one if it doesn't fall over and still needs removing because only the bravest lumberjack will climb a half-dead tree.
I am assuming you are not in an area with tree preservation by-laws - if I am wrong or you are unsure, you should contact your local council as there can be big penalties if you get it wrong.
If there are no rules about tree removal, then I would suggest you contact your local council - or another in the area - or the DPI for your state, as they should have arborists who can give you some information to help you make the safest choices and also protect whatever wildlife you may disturb in the process.

I second Alison's comments. Ring barking is not advisable as it causes an even bigger hazard. She makes a good point, an arborist would not want to climb a dead tree to remove it.

If the tree is in a forested area, then I suppose that is a different matter as there are many trees that die naturally in a forest and provide homes for animals whilst doing so.

Why would you want to ring bark the tree, anyway ?

I'd have been interested in the OP's situation too.

Escaped torellianas and excess lemon scenteds in the forestry plot do get ringbarked in my acreage situation. Like using the old alternative to herbicide, and don't the birds love a outlook!