Voldemort has a habit of telling his pet snake Nagini to keep close to him, presumably so that Voldemort can protect him. If this is the case, why didn't Voldemort simply wear the Horcrux ring (Marvolo Gaunt's Ring)? Wouldn't a ring worn always on the finger be much easier to protect than a dangerous-looking snake that humans want to kill? Furthermore, that ring never dies biologically like a snake. In fact, why doesn't Voldemort simply keep all his horcruxes near himself for easier protection given that he is so powerful? His enemies do not stand a chance against him in a one-to-one fight.
On the night Voldemort attacked the Potters and the spell rebounded on him his body was destroyed. Dumbledore would have had easy access to all of his horcruxes if he every got killed accidentally.
What does this mean?
Voldemort forsaw that his physical body could die by happenstance. So the most logical conclusion for him to make sure he could never be defeated was to hide the fragments of his soul. It was practically a miracle that Harry and Co. managed to find all of the piece.
A far better place to protection would be perhaps the bottom of the ocean, however that was not inline with Voldemorts MO of keeping trophies of his conquests, which also correlate to his locations for horcruxes.
‘And they could be anything?’ said Harry. ‘They could be old tin cans, or, I dunno, empty potion bottles ...?’ ‘You are thinking of Portkeys, Harry, which must be ordin- ary objects, easy to overlook. But Lord Voldemort use tin cans or old potion bottles to guard his own precious soul? You are forgetting what I have shown you. Lord Voldemort liked to collect trophies, and he preferred objects with a powerful magical history. His pride, his belief in his own superiority, his determination to carve for himself a startling place in magical history; these things suggest to me that Voldemort would have chosen his Horcruxes with some care, favouring objects worthy of the honour.’
Further reading on this topic Why didn't Voldemort hide his Horcruxes in impossible-to-reach places?.
Voldemort may have known the stone's power
Dumbledore's momentary lapse in judgement with the ring is out of character, which suggests that the stone may compel the holder to use it:
'When I discovered it, after all those years, buried in the abandoned home of the Gaunts, the Hallow I had craved most of all --- though in my youth I had wanted it for very different reasons --- I lost my head, Harry. I quite forgot that it was now a Horcrux, that the ring was sure to carry a curse. I picked it up, and I put it on, and for a second I imagined that I was about to see Ariana, and my mother, and my father, and to tell them how very, very sorry I was ...'
(Harry Potter and the Deathly Hallows, Chapter 35: King's Cross)
Although Harry's experience with the ring was peaceful, his first act upon holding it was to activate it:
[Harry] closed his eyes, and turned the stone over in his hand, three times.
(Harry Potter and the Deathly Hallows, Chapter 34: The Forest Again)
The stone may have similarly compelled Voldemort to use it when he first held it.
Dumbledore spoke of Voldemort's fear of the dead in Half-Blood Prince and again in Deathly Hallows:
'There is nothing to be feared from a body, Harry, any more than there is anything to be feared from the darkness. Lord Voldemort, who of course secretly fears both, disagrees.'
(Harry Potter and the Half-Blood Prince, Chapter 26: The Cave)
'And Voldemort never knew about the Hallows?'
'I do not think so, because he did not recognise the Resurrection Stone he turned into a Horcrux. But even if he had known about them, Harry, I doubt that he would have been interested in any except the first. He would not think that he needed the Cloak, and, as for the stone, whom would he want to bring back from the dead? He fears the dead. He does not love.'
(Harry Potter and the Deathly Hallows, Chapter 35: King's Cross)
If Voldemort had any experience --- or even any suspicions --- about the ring's true power and the ease with which it could be activated, then he would not have risked keeping it with him.
The ring represented Voldemort's faded heritage
Although Voldemort was proud of his ancient Peverell heritage and treasured the ring for its link to past family glory, the ring was also a reminder of his more recent ties to the Gaunt family. By the time Voldemort acquired the ring (Half-Blood Prince, Ch.17: A Sluggish Memory), he had lost all respect for his mother, and his encounter with his uncle at the family home did nothing to make him a proud Gaunt. By burying the ring in the ruins of the Gaunt shack, Voldemort may have been symbolically burying his recent family history.
He didn't want to.
In Chapter Twenty-Three of Half-Blood Prince Dumbledore discusses Voldemort's decision to hide the ring in the Gaunt shack (my emphasis):
"Well, as you now know, for many years I have made it my business to discover as much as I can about Voldemort's past life. I have traveled widely, visiting those places he once knew. I stumbled across the ring hidden in the ruin of the Gaunt’s house. It seems that once Voldemort had succeeded in sealing a piece of his soul in side it, he did not want to wear it anymore. He hid it, protected by many powerful enchantments, in the shack where his ancestors had once lived (Morfin having been carted off to Azkaban, of course), never guessing that I might one day take the trouble to visit the ruin, or that I might be keeping an eye open for traces of magical concealment.
Here Dumbledore suggests that Voldemort didn't want to wear it anymore, though he doesn't suggest a reason why Voldemort wouldn't have wanted to wear it. And he also notes that Voldemort thought that the ring would be perfectly safe even if not kept on his person, so to his mind there would have been no need to wear it.
It is perhaps also possible that Voldemort didn't want to keep the ring because it could have attracted too much attention. As per Chapter Seventeen of Half-Blood Prince the Ministry was apparently aware that there was a missing ring connected to the Gaunts:
"So the Ministry called upon Morfin. They did not need to question him, to use Veritaserum or Legilimency. He admitted to the murder on the spot, giving details only the murderer could know. He was proud, he said, to have killed the Muggles, had been awaiting his chance all these years. He handed over his wand, which was proved at once to have been used to kill the Riddles. And he permitted himself to be led off to Azkaban without a fight.
All that disturbed him was the fact that his fathers ring had disappeared. 'He'll kill me for losing it,' he told his captors over and over again. 'He'll kill me for losing his ring.' And that, apparently, was all he ever said again. He lived out the remainder of his life in Azkaban, lamenting the loss of Marvolo's last heirloom, and is buried beside the prison, alongside the other poor souls who have expired within its walls."
At some point Voldemort might have realized that keeping the ring could connect him to the Gaunts and the murders that he had committed. To stay on the safe side he might have then decided to store the ring elsewhere.