Renter's insurance covers the value of your personal items, not the physical structure of a building. A broken door frame is typically the landlord or owner's responsibility. While his homeowner's policy may cover the damage, it is often more sensible for him just to pay for the replacement himself because of a deductible on the policy.
Renter's insurance policies do normally include liability protection if someone else is injured in your home or if you damage their property, meaning it won't pay the costs of repairing a door frame that you broke. You can check with the insurer about extended coverage options for damage you cause to the physical property. The landlord may require you by contract to pay for the broken door frame if you caused the damage.
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