Is it from the authentic sunnah to say to another muslim in Ramadan, "Ramadan Mubarak?" I've heard people say "Ramadan Mubarak" and they'd reply "Ramadan Karim" or even saying "Eid Mubarak" and one replying "Eid Karim." Lastly, I've heard people say "Ramadan Mubarak" and getting the reply "Ramadan Mubarak." Is this from the innovation of splitting the month of Ramadan into three parts from its beginning-middle-and end etc. or are these sayings from the authentic athaar and is there a reward for saying these words?
Abu Bakr Al-Siddiq Radhiyallahu anhu said: I will not leave anything Allah's Messenger Sallahu alihi wa ssallam did, except that I will also do it; for I fear that if I were to leave any of his commands and ways I would deviate.(Al-Bukhari, Al-Sahih 2:386