On Petrine Christianity, maybe you are referring to Petrine Primacy?
A number of Christian denominations and scholars hold that Simon Peter was the most prominent of the apostles, favored by Jesus of Nazareth with the first place of honor and authority.[1] This doctrine is known as the Primacy of Simon Peter or the Petrine Primacy (from the Latin Petrus=Peter). A number of traditions, most notably Roman Catholic, hold that Simon Peter, also known as Saint Peter or Cephas, was the first Bishop of Rome and was a martyr during the persecution of the emperor Nero. The early Christian church at this time, however, did not have precisely delineated functions for bishops or other official roles.[2].. Primacy of Peter; Wikipedia