What is the relation between the success of marriage between the amount of money paid for rings?

There is no correlation. My dad (50 yrs married and counting) bought an expensive ring, I bought a more modest ring (23 yrs and counting), my brother bought a very cheap ring (since replaced with a much more expensive ring) and is 15 yrs married and counting. What matters to a relationship is commitment not the value of the ring you buy. If that is your concern then your marriage is doomed to failure.