To directly answer your question, short and sweet, <PERSON> is clearly not invincible even with the One Ring. During the Battle of Mount Doom (I believe that's what it was called), the combined efforts of <PERSON>, <PERSON> and <PERSON> defeated <PERSON> while he was wearing the One Ring.
As previous posts have explained, <PERSON> is a Maia, a sort of deity-like immortal spirit. As <PERSON> had explained to <PERSON>, destroying the Ring wouldn't technically kill <PERSON>, rather it would reduce him to shadow in the dark gnawing at himself in the Abyss. So to you or I in Middle Earth terms, dead for all intents and purposes.
But besides destroying the Ring, what could defeat <PERSON>? Well <PERSON> is a Mayia, and said to be of the highest order of Maia (<PERSON>, the other wizards and <PERSON> are Maiar as well). But again, he is only one Maia of many. 773
Also, there are the Valar, to which the Maiar are subservient in class and power (you might say the Valar are the equivalent of Greek or Norse Gods). So what I'm getting at is that Ring or no Ring, the combined might of the Valar and Maiar could easily defeat <PERSON>.
So why go on a Ring quest when all of the Ainur (Valar and Maiar) can easily deal with <PERSON>? Well they actually had done just that against the First Dark Lord, <PERSON> at the end of the First age. This brought about damage and upheaval to Middle Earth of apocalyptic proportions. So to avoid this fiasco again and to defeat <PERSON> in Middle Earth, the Valar send the Istari, five Maiar in limited (old man) forms to Middle Earth that we later come to know to be the Wizards. I assume the idea behind the limited forms serves as both initially a disguise from <PERSON> and to limit collateral damage in any confrontation.
There might have been other ways for the Wizards to defeat <PERSON>, but centuries of work on their part (well mainly <PERSON> and <PERSON>) show that finding and destroying the Ring to be the easiest "<PERSON>' Heel" approach. 88