Just like Fache's crux gemmata's important to establishing the character's beliefs and priorities, Aringarosa's bishop ring carries some significance as well. (These two supporting characters have better jewelry than basically anyone else in the novel.)
He takes a lot of pride in his role in the Church, especially as one of the leaders of the sect known as Opus Dei. Before he learns that the new pope is removing his prelature-status, he feels like he's dedicated his life to something with purpose and meaning. So, by wearing a "14-karat gold bishop's ring with purple amethyst, large diamonds, and hand-tooled mitre-crozier appliqué" (5.2), he's adorned with a symbol of his power and status.
Once the fate of his church is revealed to him, his ring becomes an almost painful reminder, but at the same time it is a hint of what is to come:
Trying to ease his nerves, the bishop meditated on the purple amethyst in his ring. Feeling the textures of the mitre-crozier appliqué and the facets of the diamonds, he reminded himself that this ring was a symbol of power far less than that which he would soon attain. (34.26)
Then, as things get more and more desperate, his ring becomes a bargaining chip. When he's trying to convince the pilot to take him to London, he uses it to bribe the unwilling chap:
The pilot eyed the bishop's gold ring. "Real diamonds?"
Aringarosa looked at the ring. "I could not possibly part with this."
The pilot shrugged, turning and focusing back out the windshield.
Aringarosa felt a deepening sadness. He looked at the ring. Everything it represented was about to be lost to the bishop anyway. After a long moment, he slid the ring from his finger and placed it gently on the instrument panel. (75.19-22)
Everything it represented at the time was his power within the church, and since that's being threatened it's no big whoop, right? Actually, the ring means more to him than he's willing to admit. As Brown starts to wrap up this tale of intrigue, we get some happy closure for Aringarosa, the Teacher's unfortunate pawn:
Several moments passed. Finally, Fache ran his hand over his forehead, slicking back his hair as he gazed down at Aringarosa. "My Lord, before I return to Paris, there is one final matter I'd like to discuss— your impromptu flight to London. You bribed a pilot to change course. In doing so, you broke a number of international laws."
Aringarosa slumped. "I was desperate."
"Yes. As was the pilot when my men interrogated him." Fache reached in his pocket and produced a purple amethyst ring with a familiar hand-tooled mitre-crozier appliqué.
Aringarosa felt tears welling as he accepted the ring and slipped it back on his finger. "You've been so kind." He held out his hand and clasped Fache's. "Thank you." (103.29-32)
Even though it's lost its power as a symbol of authority and leadership, its value is now more sentimental in nature. Aww. Sophie gets reunited with her fam, and Aringarosa gets reunited with his bling. All's well in the world.