1. Since it's transferred via Will, it's self acquired.
2. She has no rights in this property. You are lucky as she has moved out of the house/property, if not she could have claimed residential rights. But having said that, she has all the right to stay with you where ever you are residing. Suppose you are residing in this property, then she can very well come and stay there. This doesn't mean you have to move out now itself. Suppose a day comes when she comes back to stay with you, then you can make a separate house and then she cannot say that she wants to stay in this property.
3. Since it's a DV case, it's not very serious. Better go ahead and file a divorce.
4. She will to a greater extent say she doesn't want a divorce. If she wants to come back, as mentioned above make a separate rented house, and stay there till she settles. After that you need not go to that rented house (But you will have to pay the rent). She cannot claim the property, only thing she can claim is residence right to stay where ever you are. If you stay in a rented house, then she will have to stay there itself. And after a while when you stop staying (don't vacate it fully) in that house, even the court cannot ask as why you are not going to home and staying/sleeping with her.
In the mean time when you file for a divorce, you can obviously say that due to mental cruelty you are staying away from her.
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