No, Peter did not "rule" the Church from Rome. What he and Paul did was to lend their authority to the see of Rome, especially since each man was martyred there. As a result the capital city of the empire became the main focal point of the Church, especially after the destruction of Jerusalem.
Well, shucks. I thought Peter was the see of Rome.
"What he and Paul did was to lend their authority to the see of Rome, especially since each man was martyred there. As a result the capital city of the empire became the main focal point of the Church, especially after the destruction of Jerusalem."
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What proof is there that Peter was martyred upside down in Rome. We know Paul was because of the books he wrote.
I think your second point is much more on target about why Rome emerged as the dominant power within the early church. They were located where the money and power of the state was.