The One Ring's most important function was as an anchor to the physical world. Infusing much of himself into an all-but-indestructible object guaranteed Sauron's ability to regenerate if his body was slain. I also theorize that it acted as a kind of reservoir that he could put power into, then regenerate, and repeat, providing a way to increase the amount of power he could call upon beyond his natural limit. Other than this, the One Ring didn't do much. Enslaving bearers of the Rings of Power yielded very limited results, and I would debate whether the Ring actually granted Sauron invisibility as it did to the lesser bearers.
What does this have to do with the topic? Saruman knew he was incarnate, and doubtless knew that death as an incarnate would be permanent. The only way out for the Istari was to be restored by the Valar who had sent them, but Saruman had betrayed them. So what could he do? He probably discovered the true genius of having such a ring as Sauron's, and wanted an anchor and bank vault for himself. Obviously he didn't achieve what he was going for, but in time he may have.
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