Yes a trust can own a business.
If it is a sole proprietor, the trust will file a trade or assumed name at the government unit in your state. It would be styled as:
The Bob Smith Living Trust
Dba Floral Enterprises.
We also see trusts as owners of other types of businesses such a single member LLC. If the trust is a single-member owner it may look very similar to the dba and styled as:
The Bob Smith Living Trust
Floral Enterprises, LLC
In this case the trust is the tax reporting entity. This is called disregarding the LLC for tax purposes.
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Learn more about Deborah Crawford’s webinar Business Accounts—Beyond the Basics