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This depends on the nature of your rental.
1) If you have a LEASE, then the landlord cannot ask you to have insurance unless the lease mandates it.
2) If you are a month to month tenant, then:
(a) if you had a lease in the beginning, then its rules apply and then see #1. But,
(b) if you did not have a lease ever, then, the landlord can ask that this be a requirement with a month's notice.
Finally, you cannot insure the home as the owner. However, you can insure it as a renter - renter's insurance. This is what I mean by the answer above. Someone in your situation simply cannot insure a home as an owner if you do not in fact own it so if this is what the landlord is asking, they are asking for something improper and this cannot (or should not) be done.
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