It's explained in Game of Thrones Season 5:
Inside the Episode #10 .
Start watching from 4:50 to get your answer.
In short: The ring is "bread crumb" to point the direction that she's been taken.
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Why the tv-script writers thought she should drop her ring, who knows.
It does sound like Nika's answer is the official story.
I would like to add some things, and a few thoughts.
It does seem like this is an homage to Lord of the Rings, when Merry and Pippin are abducted by orcs, and leave a piece of jewellery behind as a way to signal their friends.
GRRM is fond of homages, and his books are full of them.
Not idly do the leaves of Lorien fall.
If this is the case, it is rather weak, in my opinion.
Aragorn and Legolas had extraordinary abilities, and followed a given track.
This was in the middle of nowhere.
If she expects anyone to find it, she is optimistic on the verge of delusional.
If this is an actual plot line that is going to happen, it is rather far fetched.
Another explanation might be that she would get rid of evidence of her new marriage to Hizdahr, or her old marriage to Khal Drogo.
However, seeing as she was discovered because she was flying a dragon, the chance of any such ruse to work would be microscopic indeed.
"
Say, perhaps she is some other girl riding a dragon.
"
No.
However...
SPOILER Season 6, and/or
ADWD/TWOW -- Mystery spoiler, be warned! as the script writers are not unaware, this is when Mirri Maaz Duur's prophecy starts to come true: "When the sun rises in the west and sets in the east," said Mirri Maz Duur.
"When the seas go dry and mountains blow in the wind like leaves.
When your womb quickens again, and you bear a living child.
Then he [Drogo] will return, and not before."
As described here , the following things have happened in the books: - Dorne has "risen" in Westeros, and Quentyn died in Mereen, "The Suns Son" -
The Dothraki sea -- the vast grasslands around Mereen -- have gone dry with the coming winter - Rhaegal and Viserion broke free and destroyed pyramids (mountains) in Mereen - Daenerys' womb quickened when she had her "moon blood" (menstruation)
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Not idly do the leaves of Lorien
fall
she was leaving something for jorah and daario to track.
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No... she dropped the ring because this is the original army she was with when she was married to drogo... and according to their tradition... she couldn't get married again.
hence why she dropped the ring
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So who was this army that immediately surrounded our new favorite dynamic duo?
If the books are any indication, they are none other than a khalasar Dothraki army, led by Khal Jhaqo, who was a former lieutenant of Drogo’s before his untimely death.
So, naturally, this is bound to make things a little awkward for all parties involved, given that a Khaleesi is traditionally supposed to live with the crones of the Dosh Khaleen shortly after her Khal dies.
Since Dany failed to do so, they may be a little curious (and more than a little upset) if they catch wind of what she’s been up to instead.
Like the fact that she got married again to a nobleman of Meereen, Hizdahr zo Loraq.
Hence why she dropped her ring out of sight as the army approached her.
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In the episode of Game of Thrones "Mother's Mercy" (S5 E10) when    Daenerys is surrounded by a Khalasar,    she drops a ring into the dirt.  Do we know the significance of this ring? Is it meaningful that she dropped that particular piece of jewelry into the dirt?
Why did Daenerys drop her ring?
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 The ring may be an indicator of the direction that she has been taken.  Another explanation might be that she is getting rid of evidence of her new marriage to Hizdahr, or her old marriage to Khal Drogo. The reason Daenerys drops the ring is either to help anyone searching for her, including her friends, to find her or to get rid of any evidence she is or was married. However, the fact she drops the ring into dirt is not significant. 
Daenerys drops the ring so that anyone looking for her, including her friends, knows where to find her. Daenerys drops the ring to erase any evidence of her last marriage to Drogo and/or her current marriage to Hizdahr. It is not meaningful that Daenerys drops her ring into dirt per se.