The corporation continues as a separate legal entity. The business assets and liabilities belong to the corporation itself, not its owner. His shares would be passed to his next of kin or, if he didn't have one, would become property of the state and most likely auctioned off.
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As with any asset, it depends on what the will says and what the estate can do.
The business should also have continuity plans for when people in important positions die.
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