Question: My body leaves; my core can stay,  

Where I shall breathe another way.  

The route of south I always linger.  

My arms have arms; my ring no finger.

What am I?

Answer: 
Are you a:

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> tree?
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My body leaves; my core can stay,

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> A double meaning: The body is leafs/leaves, and when they go away for the winter, the trunk/core remains
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Where I shall breathe another way.

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>  With no leaves, a tree can still exchange oxygen and CO2, just in a more dormant state
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The route of south I always linger.

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>  Route/root is another double meaning. [Via OP: roots stretch downward, and are harder to move the deeper they go] 
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My arms have arms; my ring no finger.

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>  The arms are branches, which can have more branches stemming from them. Tree trunks can have rings, but do not have fingers.
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Are you a 

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> Black hole ?
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Because :

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> A black hole is the core of a former star, whose most of its matter was expulsed through space via a supernova.
> Black holes are often surrounded by rings of hot gas. No idea for the arms or the south hints though.
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