You do not have to register your trademark. You own your trademark as soon as you start using it with your goods or services. However, federal registration gives you stronger intellectual property protections, so it is worth considering. This USPTO page discusses in greater depth why you might want to register your trademark. If you're interested in registering your trademark, this Trademark Registration Kit created by the USPTO walks you through the entire process with links to further resources.
Please reach out to the PTRC Representatives if you would like additional guidance. We can be contacted by email at lib-ptrc@osu.edu.
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