Let be a principal ideal domain and let be a torsion unital left -module. If is a prime in , the -primary component of is the set .
- Prove that is a submodule of .
- Prove that this definition of -primary component agrees with the definition given in this previous exercise if has a nonzero annihilator.
- If is the set of all primes in (up to associates), prove that . (Where denotes an internal direct sum.)
- It is clear that . Note also that if and , then . Hence . That is, is a chain of submodules of . Thus is a submodule of .
- Suppose is nonzero; say . In this previous exercise we defined the -primary component of to be . Certainly . Now suppose ; say . Since and is a PID, is in where t \leq k_i$. So , and so . Thus if has a nonzero annihilator, the two notions of -primary component agree.
- Now let be a finite set of primes in , and fix some . Suppose . Say , where and for each . Since , we have and (for each such that and . Note that and . Now . Since is a PID and and are relatively prime (as we know their factorizations), . In particular, . Thus . Moreover, note that if , then for some since is torsion. Say . Let . Note that , so that for some . Finally, , so that . Finally, . Thus , by this characterization of arbitrary internal direct sums.