if divorced and give vehicle in my name to ex can they carry insurance on it and myself not responsible or liable

If you have given a vehicle to your ex-spouse, they can carry insurance on it and you will not be responsible or liable for the coverage, but only if the vehicle's ownership has been legally transferred to your ex-spouse's name. If you are still listed as the legal owner of the vehicle, you could still be held liable for any accidents or damages that occur, even if your ex-spouse has insurance coverage. To ensure that you are not held liable for any accidents or damages involving the vehicle, you should transfer ownership of the vehicle to your ex-spouse's name. This process may involve filling out paperwork with the Department of Motor Vehicles (DMV) in your state and paying any associated fees. Once the ownership has been transferred, your ex-spouse can obtain their own insurance policy to cover the vehicle. However, you should also contact your insurance company to remove the vehicle from your policy to avoid any confusion or potential liability issues.