There are many words to describe someone who always needs to be right, including indomitable, adamant, unrelenting, insistent, intransigent, obdurate, unshakeable, dictatorial.
To convey more an implicationthe sense of getting angry when disagreed with, strident or truculent. - eager or quick to argue or fight; aggressively defiant, bad-tempered and always willing to argue with people: a truculent attitude.