Let be a principal ideal domain which is not a field. Prove that no finitely generated -module is injective.
Let be a finitely generated -module. By Theorem 5 (FTFGMPID), we have for some . By this previous exercise, is injective if and only if and are injective. We claim that if is injective, then .
We begin with a lemma.
Lemma: If is a finitely generated torsion module over a principal ideal domain, then there exists a nonzero element such that . Proof: By FTFGMPID, we have for some primes and nonnegative natural numbers . Now is nonzero and clearly annihilates .
By Proposition 36 on page 396 of D&F, (if nontrivial) cannot be injective, since we have for some . So if is injective, then .
Consider now ; if , it suffices to consider a single copy of . If is not a unit, then . So again by Proposition 36 on page 396, is not injective, so that is not injective if .
So if is injective, then .
Conversely, the zero module is trivially injective since every short exact sequence splits (trivially).
So over a principal ideal domain which is not a field, no nontrivial module is injective.