[Maédhros]

Forgive me if this turns into a debate over semantics, but that reads as if Sauron did not need the ring to rehouse. Are you speculating that, after the Numenor incident, the ring was needed to keep his hroa in ME after the Last Alliance defeat?

My point was that w/o the ring, Sauron would have been able to rehouse after Numenor - but only as a spirit of malice, akin to as he was in LOTR. Thus a physical struggle at the Last Alliance would have been impossible.

Certainly in the story of Lay of Leithian Sauron's (Thû) hröa was destroyed and he was able to reclothe himself again, without the need for the Ring.
But the bigger question is: What is the Ring made of? Does the Ring has part of Sauron's "spirit" (power) in it?
I don't think that I'm explaining myself correclty to you. I'm not sayin that the Ring was needed to keep his hröa in ME. When his hröa is destroyed it remains destroyed, Sauron needs to use more of his power to reclothe himself. As this process repeats himself, Saruon looses more of his innate powers, ex. Balrogs who once destroyed couldn't reclothe themselves.

This is what I believe would have happened sooner to Sauron if he had not made the Ring:
From Morgoth's Ring: Myths Transformed

The war was successful, and ruin was limited to the small (if beautiful) region of Beleriand. Morgoth was thus actually made captive in physical form, and in that form taken as a mere criminal to Aman and delivered to Námo Mandos as judge – and executioner. He was judged, and eventually taken out of the Blessed Realm and executed: that is killed like one of the Incarnates. It was then made plain (though it must have been understood beforehand by Manwë and Námo) that, though he had 'disseminated' his power (his evil and possessive and rebellious will) far and wide into the matter of Arda, he had lost direct control of this, and all that 'he', as a surviving remnant of integral being, retained as 'himself and under control was the terribly shrunken and reduced spirit that inhabited his self-imposed (but now beloved) body. When that body was destroyed he was weak and utterly 'houseless', and for that time at a loss and 'unanchored' as it were. We read that he was then thrust out into the Void. That should mean that he was put outside Time and Space, outside Eä altogether; but if that were so this would imply a direct intervention of Eru (with or without supplication of the Valar). It may however refer inaccurately* to the extrusion or flight of his spirit from Arda.
In any case, in seeking to absorb or rather to infiltrate himself throughout 'matter', what was then left of him was no longer powerful enough to reclothe itself. (It would now remain fixed in the desire to do so: there was no 'repentance' or possibility of it: Melkor had abandoned for ever all 'spiritual' ambitions, and existed almost solely as a desire to possess and dominate matter, and Arda in particular.) At least it could not yet reclothe itself. We need not suppose that Manwë was deluded into supposing that this had been a war to end war, or even to end Melkor. Melkor was not Sauron. We speak of him being 'weakened, shrunken, reduced'; but this is in comparison with the great Valar. He had been a being of immense potency and life. The Elves certainly held and taught that fear or 'spirits' may grow of their own life (independently of the body), even as they may be hurt and healed, be diminished and renewed. The dark spirit of Melkor's 'remainder' might be expected, therefore, eventually and after long ages to increase again, even (as some held) to draw back into itself some of its formerly dissipated power. It would do this (even if Sauron could not) because of its relative greatnes

It is my personal opinion but I think that without the Ring, Sauron would not have been able to be renew again and would have been a no factor in the TA.

The following was added marginally after the page was written:
If they do not sink below a certain level. Since no fëa can be annihilated, reduced to zero or not-existing, it is no[t] clear what is meant. Thus Sauron was said to have fallen below the point of ever recovering, though he had previously recovered. What is probably meant is that a 'wicked' spirit becomes fixed in a certain desire or ambition, and if it cannot repent then this desire becomes virtually its whole being. But the desire may be wholly beyond the weakness it has fallen to, and it will then be unable to withdraw its attention from the unobtainable desire, even to attend to itself. It will then remain for ever in impotent desire or memory of desire.