I took on an existing lease when I bought my latest rental unit

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I took on an existing lease when I bought my latest rental unit. I will keep the tenant and the property management company that the previous owner used. Is it normal to sign a new lease with an existing tenant or just to take over the existing lease without signing anything new?

My concern is the current lease has the previous owner's name and signature on it. My name is not on it at all. Its not that I want to change any of the terms and conditions in the lease but what legal rights will I have if I ever need to evict the tenant, for example? When I mentioned this to the property management company they clearly thought my concern was strange and that by the mere fact of my taking ownership of the property, all rights in the lease were automatically assigned to me.

What would you do? Would you insist on signing a new lease or, at a minimum, an addendum to the lease? Would you go a step further and request that the previous owner also sign a release that they relinquish all rights under the lease agreement in my favor?