Peter, as an apostle, had no successor in his office; nor was he bishop of Rome.
You are wrong. You have been misinformed. I once believed as you do, but I formed my opinion without first learning the facts.
"And thus preaching through countries and cities, they appointed the first-fruits [of their labours], having first proved them by the Spirit, to be bishops and deacons of those who should afterwards believe. Nor was this any new thing, since indeed many ages before it was written concerning bishops and deacons. For thus saith the Scripture a certain place, 'I will appoint their bishops s in righteousness, and their deacons in faith.'...
Our apostles also knew, through our Lord Jesus Christ, and there would be strife on account of the office of the episcopate. For this reason, therefore, inasmuch as they had obtained a perfect fore-knowledge of this, they appointed those [ministers] already mentioned, and afterwards gave instructions, that when these should fall asleep, other approved men should succeed them in their ministry...For our sin will not be small, if we eject from the episcopate those who have blamelessly and holily fulfilled its duties." Pope Clement, Epistle to Corinthians, 42, 44 (A.D. 98).
"For what is the bishop but one who beyond all others possesses all power and authority, so far as it is possible for a man to possess it, who according to his ability has been made an imitator of the Christ off God? And what is the presbytery but a sacred assembly, the counselors and assessors of the bishop? And what are the deacons but imitators of the angelic powers, fulfilling a pure and blameless ministry unto him, as Anencletus and Clement to Peter?" Ignatius, To the Trallians, 7 (A.D. 110).
'You have thus by such an admonition bound together
the plantings of Peter and Paul at Rome and Corinth." Dionysius of Corinth, Epistle to Pope Soter, fragment in Eusebius' Church History, II:25 (c. A.D. 178).
"Matthew also issued a written Gospel among the Hebrews in their own dialect,
while Peter and Paul were preaching at Rome, and laying the foundations of the Church." Irenaeus, Against Heresies, 3:1:1 (c. A.D. 180).
"The church of God which sojourns at Rome to the church of God which sojourns at Corinth ... But if any disobey the words spoken by him through us, let them know that they will involve themselves in transgression and in no small danger." Clement of Rome, Pope, 1st Epistle to the Corinthians, 1,59:1 (c. A.D. 96).
"And he says to him again after the resurrection, 'Feed my sheep.' It is on him that he builds the Church, and to him that he entrusts the sheep to feed. And although he assigns a like power to all the apostles, yet he founded a single Chair, thus establishing by his own authority the source and hallmark of the (Church's) oneness.
No doubt the others were all that Peter was, but a primacy is given to Peter, and it is (thus) made clear that there is but one flock which is to be fed by all the apostles in common accord. If a man does not hold fast to this oneness of Peter, does he imagine that he still holds the faith? If he deserts the Chair of Peter upon whom the Church was built, has he still confidence that he is in the Church? This unity firmly should we hold and maintain, especially we bishops, presiding in the Church, in order that we may approve the episcopate itself to be the one and undivided." Cyprian, The Unity of the Church, 4-5 (A.D. 251-256).