If Frodo took the Ring for his own then perhaps the Ring's Power would become the main source of control in Frodo's mind, and Frodo might only have become a mere 'host' for the Ring. Two possibilities then arise. He could either have chosen to try and become a dominant Power himself and fight Sauron (there seems to be more of Sauron's power in the Ring than whats left in Sauron himself by this time). Or otherwise the Ring might have simply forced Frodo to give the Ring to Sauron.

    Another interesting and engaging thread!  God I love this place!
    Before I leap out onto this alternative history of The Ring I would like to suggest that Gollums fall into lava of Mt. Doom is foretold.
    
    In The Two Towers, (Book IV: Chapter 1, The Taming of Sméagol), Gollum is begging  Frodo to remove a rope tied to his ankle.  Frodo is looking for a way to foreswear Gollum.  Gollum himself suggests  swearing on the One Ring.  Frodo is incensed.
     Frodo drew himself up, and again Sam was startled by his words and his stern voice, On the Precious? How dare you? he said. Think! One Ring to rule them all and in the Darkness bind them.  Would you commit your promise to that Sméagol?  It will hold you.  But it is more treacherous than you are.  It may twist your words  Beware! (The Two Towers, Book IV: Chapter 1, The Taming of Sméagol, pg 265),
    So Gollum swears by it (not on it) and Frodo has Sam untie the rope. And the three begin their journey to the Black Gate.
    A couple of chapters later, when the trio is at the Black Gate, Gollum is terrified at Frodos plan to enter Mordor.  He is worried about Sauron capturing The One Ring. He proposes another path but Frodo is unsure.  But in the end he agrees to follow Gollum.  But mindful of the creatures treachery he speaks,
    Sméagol, he said, I will trust you once more.  Indeed it seems that I must do so, and that is my fate to receive help from you, where I least looked for it, and your fate to help me whom you long pursued with evil purpose.  So far you have deserved well of me and have kept your promise truly.  Truly I say and mean, with a glance at Sam, for twice now we have been in your power, and you have done no harm to us.  Nor have you tried to take from me what you once sought.  May the third time prove the best!  But I warn you, Sméagol, you are in danger.
    Yes, yes master! said Gollum. Dreadful danger!  Sméagols bones shake to think of it, but he doesnt run away.  He must help nice master.
    I did not mean the danger that we all share, said Frodo, I mean a danger to yourself alone.  You swore a promise by what you call the Precious.  Remember that!  It will hold you to it; but it will seek a way to twist it to your own undoing.  Already you are being twisted.  You revealed yourself to me just now, foolishly. Give it back to Sméagol, you said.  Do not say that again.  Do not let that thought grow in you!  You will never get it back.  But the desire it may betray you to a bitter end.  You will never get it back.  In the last need, Sméagol, I should put on the Precious; and the Precious mastered you long ago.  If I, wearing it, would to command you, you would obey, even if it were to leap from a precipice or to cast yourself into the fire.  And such would be my command.  So have a care, Sméagol!(The Two Towers, Book IV: Chapter 3, The Black Gate is Closed, pg. 291 -292)
    Much later on (in the Return of the King) Sam and Frodo have come to the Cracks of Mt. Doom.  Gollum appears again after a long separation.  He attacks Frodo outside of the tunnel. Frodo flings him off and rebukes him.
    Down, down! he (Frodo) gasps, clutching his hand to his breast, so that beneath the cover of his leather shirt he clasped the Ring. Down you creeping thing, and out of my path! Your time is at an end.  You cannot betray me or slay me now.
    Sam looks at Gollum and Frodo and seems to have some sort of vision in which Frodo is transformed, and before it stood stern, untouchable now by pity, a figure robed in white, but at its breast it held a wheel of fire.  Out of the fire there spoke a commanding voice.
    Begone and trouble me no more!  If you touch me ever again, you shall be cast yourself into the Fire of Doom.(The Return of the King, Book VI: Chapter 3, Mount Doom, pg. 245 -246)
    The vision passes and Sam tells Frodo to get on to destroying the Ring and that Sam will deal with Gollum.  Sam does get Gollum to flee back a little but not effectively.  He chases after Frodo and witnesses Frodo overwhelmed by the evil power of Ring, claiming the Ring for himself and not destroying it.  Gollum clunks Sam on the head with a rock and goes after Frodo and the Ring.  Frodo, now with the One Ring on, is invisible.  That doesnt phase Gollum who manages to grab Frodo and bites the Ring off Frodos hand.  Gollum is dancing around like a mad thing holding the Ring with Frodos finger still in it.  There, at the Cracks of Doom, Sammath Naur, the Ring still not possessed  by Gollum (he holds it but has not put it on) fulfills the prophesy that Frodo commanded (whose finger still possess the Ring) and Gollum goes into the Fire.
    Precious, precious, precious! Gollum cried.  My Precious! O my precious!  And with that even as his eyes were lifted up to gloat on his prize, he stepped too far, toppled, wavered for a moment on the brink, and then with a shriek he fell.  Out of the depths came his last wail precious, and he was gone. (The Return of the King, Book VI: Chapter 3, Mount Doom, pg. 249)
The Ring is destroyed.  And I think the Ring deliberately destroyed Gollum.

So to extrapolate and compose the alternate history based on the following proposal;
Frodo failed to throw the ring in. In the end, taken over by the ring itself, he refused to rid the world of the ring. If not for Gollum, the entire venture would have failed,
OK Rewrite the scene wo/Gollum.

Using the LOTR books to support the extrapolation I would like present a rational before revealing a hypothetical scenario.

First going back to Fellowship of the Ring, Book I, Chapter 1; A Long-Expected Party as Bilbo prepares to leave the Shire, he and Gandalf have a confrontation about leaving the Ring for Frodo.  Gandalf enters Bilbos home just as Bilbo is about to leave.  Bilbo has put the Ring in an envelope addressed to Frodo and sets it on the mantelpiece.  He seems to change his mind and puts it back in his pocket.  Gandalf half expects to find Bilbo invisible.  As the chapter progresses Gandalf has to exert his will to help Bilbo let go of the Ring to Frodo.  And in the end he is successful and Bilbo leaves the Ring for Frodo.
But midway through the chapter Tolkien gives us a hint at the terrible effects of  possessing the Ring long term.
    Bilbo says, I am old, Gandalf.  I dont look it, but I am beginning to feel it in my heart of hearts.  Well-preserved indeed! he snorted, Why, I feel all thin, sort of stretched, if you know what I mean: like butter that has been s****ed over too much bread.  That cant be right.  I need a change or something. (Fellowship of the Ring, Book I, Chapter 1; A Long-Expected Party, pg.54)
    Bilbo also reacts in a paranoid way towards Gandalf and in an unusual burst of temper flares out at Gandalf.
    Yes, yes, said Gandalf. But there is no need to get angry.
    If I am it is your fault, said Bilbo, It is mine, I tell you.  My own.  My precious.  Yes, my precious.
    The wizards face remained grave and attentive, and only a flicker in his deep eyes showed that he was startled and indeed alarmed. It has been called that before, he said, but not by you.
    But I say it now.  And why not?  Even if Gollum said the same once. Its not his now, but mine.  And I shall keep it, I say. (Fellowship of the Ring, Book I, Chapter 1; A Long-Expected Party,  pg. 56)
    There are more words and the paranoia rises until Gandalf has to call on his power to frighten Bilbo to let the Ring go.  And Bilbo finally does and as he lets go of the Ring his aberrant behavior subsides and, friends still, Gandalf sees him off.
    
    For purposes of the extrapolation please notice that Bilbos possession of the Ring has led to a type of behavior that is parallel to Gollum.
    Another hobbit reaction to the possession of the Ring occurs in Fellowship of the Ring, Book II, Chapter1; Many Meetings.  Frodo finally is re-united by Elrond with Bilbo in one of Rivendells halls.  Elrond leaves the hobbits together and Bilbo tells of his adventures since leaving the Shire and of the ominous warnings that Gandalf and the Elves have relayed to him regarding the Enemy and the One Ring, including the warning to not meddle with the Ring.  Bilbo of course cant resist.  He asks Frodo of the Ring.
    Have you got it here? he asked in a whisper. I cant help feeling curious, you know, after all Ive heard.  I should very much like just to peep at it again.
    Yes, Ive got it, answered Frodo, feeling a strange reluctance. It looks just the same as ever it did.
    Well, I should like to see it for a moment, said Bilbo.
    When he had dressed, Frodo found that while he slept the Ring had been hung about his neck on a new chain, light but strong.  Slowly he drew it out.  Bilbo put out his hand.  But Frodo quickly drew back the Ring.  To his distress and amazement he found he was no longer looking at Bilbo; a shadow seemed to have fallen between them, and through it he found himself eyeing a little wrinkled creature with a hungry face and bony groping hands.  He felt a desire to strike him.
    The music and singing round them seemed to falter, and a silence fell.  Bilbo looked quickly at Frodos face and passed his hand across his eyes.  I understand now, he said.  Put it away! I am sorry; sorry you have come in for this burden; sorry about everything.  Dont adventures ever have an end?.
    Frodo hid the Ring away, and the shadow passed leaving hardly a shred of memory.  The light and music of Rivendell were about him again.(Fellowship of the Ring, Book II, Chapter1; Many Meetings. Pg. 280)

    Thirdly for this purpose I bring forth the warning of Galadriel to Frodo about using the Ring
    You have not tried, she said.  Only thrice have you set the Ring upon your finger since you knew what you possessed.  Do not try!  It would destroy you.  Did not Gandalf tell you that the rings give power according to the measure of each possessor? Before you could use that power you need to become far stronger, and to train your will to the domination of others. (Fellowship of the Ring, Book II, Chapter 7; The Mirror of Galadriel , pg. 432)

    And lastly, from history of the Ring told by Gandalf to Frodo in front of the fire at Bag End comes the story of  Sméagol and his friend Déagol and their finding of the Ring.  Gandalf makes the point that these people who live near the Great River of Wilderland and not far from the Gladden Fields are very similar to hobbits.  Two members of this clan went fishing one day.  One named Sméagol got out of the boat and went exploring on the land.  The other, named Déagol, stayed in the boat to fish.  Well Déagol hooks a great fish and it pulls him out of the boat and under the water.  On the bottom he sees something shining.  It is the Ring.  He doesnt know it but he grabs it anyway.  He gets out of the water and on to the bank.  He sees its a golden ring and he feels pretty lucky.  Sméagol, who has been watching Déagol, sees the Ring and wants it.
    Give us that, Déagol, my love, said Sméagol, over his friends shoulder.
    Why? said Déagol.
    Because its my birthday, my love, and I wants it. said Sméagol.
    I dont care, said Déagol. I have given you a present already, more than I could afford.  I found this, and Im going to keep it.
     Oh, are you indeed, my love, said Sméagol; and he caught Déagol by the throat and strangled him, because the gold looked so bright and beautiful.  Then he put the ring on his finger.
    Gandalf goes on to say how Sméagol was corrupted by the Ring, driven from his family and turned into the despised creature Gollum.(Fellowship of the Ring, Book I, Chapter 2; A Shadow Of The Past,  pg. 77-78)

So what does this all bring us?
It gives us data to create a model and a scenario of what would have happened at Sammath Naur without Gollum.
First; there is obviously a mental state of violent paranoia with obsessive compulsive behaviors that are beyond the Ring wearers conscious control destroying the capacity for rational thought..
Second; we have schizophrenic affective delusions with psychotic hostile impulses, including perceptional difficulties in recognizing family members and intimate acquaintances.
Third; using the Ring triggers power and control issues in group interactions and leads to grandiose control fantasies.
Lastly, obsession over the Ring leads to actualized homicidal behaviors.
The Ring also has physical side effects that erodes stamina, and retards the natural developmental aging process.

So here is what I think would happen without Gollum;
Sam would have witnessed Frodos being overcome by his obsession for the Ring.  Being an intimate of Frodo he would attempt to reason with Frodo in terms of letting go of the Ring.  This would trigger a paranoid reaction in Frodo.  Frodos inability to see reason would trigger a delusional conception of Sam as an enemy, thereby making Sam an unrecognizable image consistent with Frodos anxiety and hallucinatory schizophrenic visions.  Sam, to Frodo, would appear as some big smelly monster whose only purpose was to not only take the Ring but also to kill Frodo.
Frodo would attempt to kill Sam.
Because of Frodos delusional state and anxiety his physical state would result in hysterical mania and unusual strength.  Sam would be in real trouble.
However Frodo would be in such psychological trauma, slipping in and out of psychosis so rapidly that, if Sam could find him, and convince him he does not want the Ring, there is a possibility that he might be able to control Frodo.  If such could happen then even mild physical restraint could trigger a response where Frodos paranoid hyperactivity could subside and he would succumb to physical and psychological exhaustion.
More likely however is the scenario that Sam would attempt to grapple with Frodo and the manic non-rational reaction would tip them both over the edge and into the fire.

Thank you for your patience.

Bear an Elf-Friend

Nice ideas Bear... I can definitely understand where you're coming from with that, however I just read letter #246 where these possible alternative endings are discussed and there is no trace of Sam there.
I would give a quote but it would a rather long one and so I simply recommend reading the letter. ;)
To sum it all up Tolkien believes that both Frodo and Gollum (had Gollum not decided to steal the Ring and had repented as he was about to before Shelob's Lair) would have eventually perceived that only evil can be done with the Ring and would have cast themselves voluntarily in the fire.
There would have been no need for Sam to do anything and as such he is never mentioned.

So if Frodo claimed the Ring either the Ringwraiths would arrive in time to escort him outside where Sauron himself would come to claim the Ring or Frodo would sacrifice himself for the sake of Middle-earth.

Hmmm...Could be... But I think they would rather enjoy a Leg-O-Las

Hi All

An old topic but what he hey !

I like the ending.
Frodo fails at the last hurdle.
It proves just how powerful Saurons ring is.

Frodo did what he could only do...his best. He is a hobbit not a Maiar or Elf or man of the west. He's just a frightend hobbit.

He claims the ring for his own and Gollum bites it off and falls and dies.

Just because you fail doesn't mean you're not a hero.

My boxing hero isn't Floyd Mayweather or Tyson or Ali.
It is Karl Taylor.
He has had around 130 fights
He has been beaten 118 times

He gets the call...packs his kit...arrives at the venue and gets beaten..but does his best. I mean it....he fights with all his heart.
He's a hero. He fails

If Gollum hadn't off bit the finger, Frodo would be toast.
Stealing a fast car does not make you a racing driver.