One of the Deadly Sins: Greed

Among the deadly sins, greed can be related to ring, from some aspects.

“Earth provides enough to satisfy every man's needs, but not every man's greed.” 
― Mohandas Karamchand Gandhi

Greed can be explained as demanding more than one's needs. Dostoyevsky linked greed and envy as:

“The world says: "You have needs -- satisfy them. You have as much right as the rich and the mighty. Don't hesitate to satisfy your needs; indeed, expand your needs and demand more." This is the worldly doctrine of today. And they believe that this is freedom. The result for the rich is isolation and suicide, for the poor, envy and murder.” 
― Fyodor Dostoyevsky, The Brothers Karamazov

Firstly, size of the diamond on the ring became a way to show "richness" so it's a kind of show-off tool. It's not for meeting a need, but for display.
Small sizes of diamonds are never thought as "enough" and even the advertisements and campaigns are based on this thought. 

“Greed, envy, sloth, pride and gluttony: these are not vices anymore. No, these are marketing tools. Lust is our way of life. Envy is just a nudge towards another sale. Even in our relationships we consume each other, each of us looking for what we can get out of the other. Our appetites are often satisfied at the expense of those around us. In a dog-eat-dog world we lose part of our humanity.” 
― Jon Foreman

Secondly, another aspect can be manufacturers' greed. Sadly they do whatever it takes to obtain the diamonds. Blood Diamond is  a movie which is about this subject:

Blood Diamond 2006
Director: Edward Zwick
http://www.imdb.com/title/tt0450259/

[The scene depicts an R.U.F. labour mining camp where captive villagers are forced to mine for diamonds.]

Captain Poison: The Freetown government and their white masters have raped your land to feed their greed! R.U.F. have freed you!

[Armed guards are shown, supervising the captives.]

Captain Poison: No more slave and master here! R.U.F. is fighting for the people! R.U.F. is FIGHTING for Sierra Leone!

[One of the workers finds a diamond and hides it in his mouth, apparently successfully.]

Captain Poison: Any bastard think he would joke with me diamond, I go cut he throat! STOP!

[Captain Poison approaches the worker.]]

Captain Poison: Give it to me!

[The worker takes the diamond out of his mouth and hands it to Poison, who then shoots him.]

References:

http://www.goodreads.com/quotes/search?utf8=%E2%9C%93&q=greed&commit=Search

http://www.diamondsourceva.com/education/caratweight/caratweight.asp

http://en.wikiquote.org/wiki/Blood_Diamond