"He can voluntarily give temporary control over a ring to his servants, however. If the servant dies or falls unconscious, the rings teleport back to the Mandarin. Conversely, if the Mandarin himself is knocked out, all the rings automatically return to him."
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"Left Ring finger — "Mental-Intensifier"

the fact, that itachi wears his ring on the right ring finger aside.

when naruto, kakashi and co were fighting that fake Itchi, didn't he also wear that ring?
and it was also shown that he could use his illusion technique on naruto with just pointing his finger on him.

so couldn't it be that they also inhabit the powers of the original wearer somehow?
or maybe the ring makes it possible that he can use his technique even when they don't look into his eyes?