Let be a principal ideal domain. Let be a left unital -module, and suppose where is nonzero. Say is the irreducible factorization of in . Prove that is the internal direct sum of its submodules of the form . (This is called the -primary component of .)
For each , let .
We claim that for all . Suppose . Note that , so that . Thus . Suppose now that . Since is a principal ideal domain and and are relatively prime, we have for some . Note then that . Thus , and we have .
Next, we claim that for all . To that end, let . Note that and are relatively prime (by definition). Now , since while . Thus as desired.
Next, we claim that . To see this, note that we have the irreducible factorization of each , and that these have no irreducible factors in common.
Now let . Since , we have for some . Now .
Thus , and in fact this sum is direct. So .