Full Idea
One of the most obvious facts about identity is that it holds one-one (John and Mr Smith) and perhaps many-many (John+Mary and Mr Smith+Miss Jones), but never one-many. It follows that composition as identity (things are their parts) is false.
Gist of Idea
Composition as identity is false, as identity is never between a single thing and many things
Source
Trenton Merricks (Objects and Persons [2003], §1.IV)
Book Reference
Merricks,Trenton: 'Objects and Persons' [OUP 2003], p.21
A Reaction
This assumes that 'having identity' and 'being identical to' are the same concept. I agree with his conclusion, but am not convinced by the argument. I'm not even quite clear why John and May can't be identical to the Smiths.