Only one name shows on my tickets from Florence to Rome. Should both passengers' names be on these tickets?

I bought a ticket, going from Florence to Rome, for me and a ticket for my wife. I printed the tickets on my home computer. Both tickets have the Train number, the coach number and the seat number on them. They both list my name as the person who purchased the ticket and they both list my name as the passenger. Should one of the tickets have my wife's name on it as the passsenger?
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