You inherit the coop at its current market value so there are no capital gains. You pay the mortgage on it from the proceeds of the sale and keep the balance.
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Inheriting a co-op is no different than inheriting shares in any other corporation. The lease is subject to co-op board approval. There is no "mortgage" but the shares are secured by a lien in accordance with the lien. If the lien is not satisfied then the shares will not be released in any attempt to sell, as your ownership interests are subject to the lien.
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Mr. Chertock & Mr. Nathonson are both correct, but I'd like to point out that your question confuses me.
You asked, "What about probate?" Did you or did you not already inherit a coop? If you inherited something, that assumes probate has already occurred. If probate has not yet occurred, you cannot have inherited anything.
I may be guessing or not licensed in your state. No atty/client relationship exists.