What is the Law on prorated rent or is there a law?

It depends on where you are. Some states address it, others don't.

Also, you really should tell us the whole story. Even if we did cite the law, it might not apply.

I was actually in search for an answer. My sister just moved to Ohio and she was told she have to pay the full rent to move in even though it will be near the end of December. I don't understand why she have to pay the full rent.

I can't find anything in the code that addresses this pro-rating. Generally, a landlord will prorate the front end if the tenant moves in prior to the lease effective date.

It is fairly simple, if the landlord will not prorate for your move in month, don't rent from them. Go find another place to rent, tell landlord that if they don't you won't rent from them