Praise be to Allah.
You did a mistake when you took the ring without permission and when you did not tell your mother after you lost it. What you have to do now is to tell your mother and ask her for forgiveness, and to compensate her with a ring like the one that she had. Because whoever takes the property of someone else without permission is a robber or one who seizes property unlawfully, and he must replace what he has taken. If this was done at a young age, below the age of adolescence, then the burden of sin is lifted, but the obligation to replace the item still remains, because the scholars have stated that a child must replace things that he has destroyed and make up for the loss in value that may have affected the things he took, then return them (to their owner).
But if you think that telling her about this will most likely lead to a breakdown in relations or problem between you, then simply give her a ring, even if it appears to be a gift.
And Allaah knows best.