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

Odour

The odour of the ring is cold and metallic:
COPPER/GOLD/SILVER/PLATINIUM
And maybe the smell of the welding process at the beginning.
Those odours don't sound like something very meaningful and emotional.
Maybe it's because the ring by itself doesn't mean much of a thing without a human being. 
It becomes warmer when a human wears it and that person's odour becomes the ring's odour.

Also there is a kind of ring that called fragrance ring:
Fragrance rings fit on top of a light bulb.  A few drops of fragrance oils are put into the groove in the ring and when the light is switched on, the heat from the bulb causes the oils to vaporise, thus providing the required fragrance of your choice. 

References:
http://www.bellasugar.com.au/New-Thierry-Mugler-Angel-Holiday-2010-Solid-Perfume-Ring-12124501
http://www.cupcakestastenice.com/archives/768

Postage Stamps

Lord of the Rings stamps:

"Make your own" custom wedding stamps

References:

http://yaymicro.com/preview.action?search.offset=35&search.id=2996675&search.limit=15&search.numResults=62472&search.searchLanguage=English&search.search=ring+stamp&search.filterExplicit=true&search.sort=relevancy&search.advanced=false

http://yaymicro.com/preview.action?search.offset=30&search.id=2006845&search.limit=15&search.numResults=63699&search.searchLanguage=English&search.search=postage+stamp+ring&search.filterExplicit=true&search.sort=relevancy&search.advanced=false

http://yaymicro.com/preview.action?search.offset=299&search.id=4463148&search.limit=15&search.numResults=63699&search.searchLanguage=English&search.search=postage+stamp+ring&search.filterExplicit=true&search.sort=relevancy&search.advanced=false

http://www.gbstampsonline.co.uk/sg2429-2438-2004-lord-of-the-rings-stamp-set-1373-p.asp

http://www.zazzle.com/engagement_ring_postage_stamp-172707409840732171

http://www.zazzle.com/platinum_love_rings_postage_stamp-172504857267043864