# Is the p-primary component of a module over a UFD canonical?

My understanding is that in general, the decomposition of a module into various forms (e.g. into cyclic modules by the elementary divisor method) is not unique because there may be non-trivial automorphisms of the module. However, some spaces are canonical, e.g. the torsion submodule. My question is: is the p-primary component of a module (i.e. the submodule of elements annihilated by some power of the prime p) canonical?

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Of course, you have just given a canonical definition of it. –  Martin Brandenburg Feb 11 at 22:51