I don't think he would hide in the mountains like Gollum. Remember that at the time that Gollum found the Ring, Sauron was not very powerful. The Ring knew this; so, It made Gollum retreat into the mountains. This was pretty much a safety measure. The Ring put itself into "storage" and decided to fall off Gollum's finger when it thought that it was the right moment. If the Belrog got the Ring, It most likely would "make" him come out into the open. That way, Sauron could easily spot it and take it for His own.
Cheers,
Mungo
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Yet the lies that Melkor...sowed in the hearts of Elves and Men are a seed that does not die and connot be destroyed, and ever and annon it sprouts anew, and will bear dark fruit unto the latest days.
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