what is the significance of the scene where portia asks for her ring?
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It is significant because Portia expected (hoped) that he's refuse, based on his promise.
Portia asks Antonio for his gloves and Bassanio for his ring, which she herself gave Bassanio on the condition that he never part with it. Bassanio pulls his hand away, calling the ring a trifle and claiming that he will not dishonor the judge by giving him such a lowly gift. Instead, Bassanio offers to find the most valuable ring in Venice, but Portia remains firm, and demands the trifle or nothing. When Bassanio admits that the ring was a gift from his wife, who made him promise never to part with it, Portia claims that the excuse is convenient and used by many men to hold onto possessions they would rather not lose. With that, she takes her leave. Antonio urges Bassanio to let the law clerk have the ring, saying that he should value Antonio’s love and the gentleman’s worth more than his wife’s orders. Bassanio gives in and sends Gratiano to run after Portia and present her with the ring.
She is testing Bassanio. Perhaps she feels the need to test him after she sees how strong his friendship to Antonio is. Notice that it is Antonio who talks Bassanio into giving up the ring after he so nobly refused to give it to "the judge."