The fixation with rings and what they represent is shown early in The Talented Mr. Ripley . In chapter one, Tom is staring right at the gold ring with the worn-out signet being worn by the senior Mr. Greenleaf when he makes the fateful decision of leaving New York. Tom...
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is no stranger to the higher classes. He knows what the rich and powerful wear and what they do not. Tom would know how to appreciate such a symbol of status, and evidence of this comes later in the novel when he becomes fixated with Dickie, his life, and his rings.
Dickie's signet ring is similar to his father's. The ring is, henceforth, a symbol of identity, position, and status. It is a show of lineage, social hierarchy, and social identity; it is distinction. No matter how bohemian and wild he claims to be, Dickie is fully aware that he is part of what could be deemed as "American aristocracy". Someone like Dickie would have made sure to show off his status as best as he knew how.
The other ring, a green one, was equally extravagant, and Dickie always wore it, just like the gold one. The rings were always under Dickie's control; they were unique to him, and nobody else could have laid hands on them, until Tom came around.
Therefore, the rings are an integral part of who Dickie is. They are essential to Dickie's personality, life history, ancestry, lineage, and status. These unique aspects of someone's life coukd be appealing to many people, but they would be especially appealing to an obsessed sociopath like Tom Ripley. When Tom decides to absorb Dickie's personality, and make it his own, part of this new identity included securing Tom's most precious items, such as his two rings.
But why keep the rings if it's dangerous? The decision of keeping the rings is no different than the decision of killing Dickie. As a sociopath, Tom cannot think past his immediate desires, so he really thinks that he can get away with making someone disappear from the face of earth and be able to cover his crime with clever answers to questions. Therefore, the boldness of killing a man, assuming his identity, manipulating his friends and family, conning people, and trying to play it all off like a big game are actions not so controversial in the mind of a delusional man.