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In organic chemistry, the word "residue" is synonymous with "moiety." These words just refer to some part of a molecule being discussed or not being discussed. Any organic [functional] group, whether it is a ketone, an ester, a vinyl group or a benzene ring can be referred to as the "ketone residue," "ester residue," "vinyl residue," etc. The word "residue" probably started being used when the organic group to which someone was referring was not part of, or was not being included in, some other part of a molecule being discussed.

When someone is referring to the "ring" of a molecule, he is almost always talking about a benzene ring, although it is possible that he could be talking about any ring regardless of how many atoms it is comprised of or whether it is aromatic or not. But in general, just the word ring usually means a benzene, or phenyl, ring - same thing.

For example, aspirin contains a benzene ring, a carboxylic acid group (attached to the ring), and an acetylated phenol moiety or group (also attached to the ring). Imagine an organic chemist is discussing with his coworkers ways to replace one of the hydrogen atoms on the methyl group with one chlorine atom. I can imagine one of the chemists saying something like "That method might work, but it might also chlorinate the ring residue."

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