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But first they would need to invent some tape
(i.imgur.com)
They could have fitted the whole ring / tape / mouse assembly into a small paper bag Aragorn could have kept it in his jacket and fed it little bits of lembas on the way how lovely x
Yes, but through that mouse, the ring would wield a power too squeak and scratch to imagine.
Mouse would escape the tape and stick its head through the ring. Then you have an invisible mouse to rule them all to deal with. The whole of Middle Earth would be absolutely overrun with mice