Symbolism, Imagery, Allegory
While it's not clear that rings have an absolutely unified symbolism in the The Duchess of Malfi, they sure do show up a lot.
The Happy Rings
The most prominent rings are the wedding rings of Antonio and the Duchess. The Duchess's is the wedding ring given to her by her first husband, the Duke of Malfi, and her putting it on Antonio's finger at 1.1.407 initiates the process of her physically raising him off of his knees and joining her as her husband and aristocratic equal (go look at "Standing and Kneeling" in this section for more on this moment).
The second time these rings show up is when, as the Cardinal is banishing the couple from Ancona, he rips off the Duchess's own wedding ring. The Cardinal vows to "sacrifice / [the ring] To his revenge" (3.4.36-37), after which the pilgrims watching the scene remark that Antonio's "own weight / Will bring him sooner to th'bottom. (3.4.40-41). Antonio's "own weight" is, of course, his own commoner lineage.
It's neat to notice that "wedding ring goes on" coincides with "Antonio rises up" and "wedding ring comes off" coincides with "Antonio sinks down." Here, it looks like the ring represents, among other things, Antonio's bold social climbing, brought about by the Duchess's marrying him.
The Less Happy, Darker Rings (Alas, Not the One Ring)
Later in the play, wedding rings get kind of twisted. When Ferdinand is torturing the Duchess, he gives her a severed hand that he tells her it's Antonio's, identified by the wedding ring it bears.
Think about it for a second. By doing this, he twists around the idea of the wedding ring, transforming it from a token of the Duchess's love for Antonio into a token of Ferdinand's revenge upon the Duchess.
And right before the Duchess is strangled, her executioner shows her the noose and says, "Here's your wedding ring" (4.2.239). The emblem of the Duchess's decision to marry Antonio is transfigured into the means by which she's murdered. How's that for irony?
The Sexy Rings. Just the One, Actually.
Rings also have a sexual undertone kind of like the one that's at play in The Merchant of Venice. The Duchess's sexuality is a huge deal, and her brothers' attempts to control her, ahem, ring, are a central part of the plot.
It's no coincidence that, when we first meet him, Antonio's fresh off a jousting tournament in France where he proved his skill with his lance and "took the ring oft'nest" (1.1.86). Penis jokes: here to prove that nothing important has changed in the last 400 years.