The office (or term of office) of a cardinal(noun)
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Use "cardinalship" in a sentence
"His canons and grand-vicars were good old men, rather vulgar like himself, walled up like him in this diocese, without exit to a cardinalship, and who resembled their bishop, with this difference, that they were finished and he was completed."
"It was in 1961, and Croetine's uncle-my great-uncle-had been appointed to a cardinalship and was then stationed at the Vatican."
"Cambrai might rely upon advancement to a cardinalship, and steps were taken, but without any good result, to bring about that event."