Hi, thank you for your question. Just a bit more information required to fully assist you:
-Was there a court approved financial settlement as part of the divorce?
-Has she remarried?
-Will she be paying rent?
I see, apologies for the confusion I thought you had meant your ex-wife. In that case, you should not enter into a tenancy agreement with her and only treat her as a lodger. Eviction of a lodger is only by reasonable notice and once the house is purchased she no longer has any legal interest or right to stay if you have given her reasonable notice to leave which I would suggest be made in writing.
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You do not necessarily need a written agreement but need to make her aware that she is only a lodger thereby not creating a tenancy. Lodgers have less rights than tenants and can easily be evicted.
There is no exact framework for reasonable notice but usually 2-4 weeks would be considered reasonable as this gives people enough time to arrange alternative accommodation.
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