What was the relationship between ancient Celtic Christianity (pre-Augustine of Cantebury) and the Bishop of Rome?
I recently read Bede's Ecclesiastical History, and now am reading several other contemporary works collected in _The Age of Bede_, and it is interesting to note the number of references to the bishop of Rome fulfilling the office of Peter, having authority over the whole Church, having the power of the keys, etc. There is a real cult of Peter, who frequently is described as "Prince of the Apostles." Rome is seen by Bede and others as having better, more truly Apostolic and orthodox traditions than say the old Irish and Briton churches, which are considered in error on many points. Indeed, in one story regarding a difference between the old traditions and the newer ones brought from Rome, the bishops are asked, who is greater, St. Columba or St. Peter, to which is exlaimed: Peter!
It's often said that before the Irish were Romanized, they had their own unique form of Christianity, closer to (Eastern) Orthodoxy, so I'm interested in seeing if their views of Rome and the Bishop of Rome were in any way different.