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While use of the copyright symbol is not required, it is a visible reminder that a given work is copyrighted and puts potential infringers on notice.
Publication is Not Required.
The U.S. Copyright Office has a definition of “publication” (Copyright Basics, p. 3). It has to do specifically with how it’s released to the public.
Registration is Not Required.
Registration is a benefit when the author of a work wishes to enforce or protect her or his copyright. Registration is a prerequisite for filing a lawsuit for copyright infringement (Copyright Basics, p. 7).
A author of a work may register a copyright with the U.S. Copyright Office by submitting an application with the appropriate fee and approved copies of the work.
Go to the U.S. Copyright Office’s Copyright Basics (p. 7) to learn how to register a copyright.