Did Salazar Slytherin know the stone in his ring was the Resurrection Stone? Or was he unaware of it's significance or just came across the stone by happenstance? In the 'tale' the second brother kills himself and it isn't mentioned whether he told others about what his stone unlike the oldest brother who boasted about the wand.
The ring never belonged to Slytherin
Gaunt only claims the locket is Slytherins, not the ring. He claims the ring as belonging to the Peverells - aka the Brothers from the story.
"That's right!" roared Gaunt. For a moment, Harry thought Gaunt was making an obscene hand gesture, but then realized that he was showing Ogden the ugly, black-stoned ring he was wearing on his middle finger, waving it before Ogden's eyes.
"See this? See this? Know what it is? Know where it came from? Centuries it's been in our family, that's how far back we go, and pure-blood all the way! Know how much I've been offered for this, with the Peverell coat of arms engraved on the stone?"
With a howl of rage, Gaunt ran toward his daughter. For a split second, Harry thought he was going to throttle her as his hand flew to her throat; next moment, he was dragging her toward Ogden by a gold chain around her neck.
"See this?" he bellowed at Ogden,shaking a heavy gold locket at him,while Merope spluttered and gasped for breath.
"I see it, I see it!" said Ogden hastily.
"Slytherins!" yelled Gaunt. "Salazar Slytherin's! We're his last living descendants, what do you say to that, eh?"
In addtition, as @b_jonas points out, this answer establishes that the Peverells came after The Founders of Hogwarts. Even if he did claim the ring was Slytherins I'd be wary of believing him - he was right about the locket but the pure blood families have a tendency to make up or exaggerate their claims to ancient ancestry. As Harry says
"Marvolo Gaunt was an ignorant old git who lived like a pig, all he cared about was his ancestry. If that ring had been passed down through the centuries, he might not have known what it really was. There were no books in that house, and trust me, he wasn’t the type to read fairy tales to his kids. He’d have loved to think the scratches on the stone were a coat of arms, because as far as he was concerned, having pure blood made you practically royal."
Or how about another example from our favourite monster?
"That’s - that’s pretty, Dolores," she said, pointing at the pendant gleaming in the ruffled folds of Umbridge’s blouse.
"What?" snapped Umbridge, glancing down. "Oh yes - an old family heirloom," she said, patting the locket lying on her large bosom. "The S stands for Selwyn.... I am related to the Selwyns.... Indeed, there are few pure-blood families to whom I am not related".