Proposition 1.2. For M ∈ M ,
M = mM =⇒ M = 0.
This can be used to give proofs of analogues of many standard results in the
theory of ﬁnitely generated modules over commutative rings. For example,
Corollary 1.3. Let M ∈ M and suppose that N ⊆ M is the image of a
morphism N −→ M in M . Then
M = N + mM =⇒ N = M.
Proof. The standard argument works here since we can form M/N in M and
as M/N = mM/N, we have M/N = 0, whence N = M.
At this point we remind the reader that over a commutative local ring, every
projective module is in fact free by a result of Kaplansky [12, theorem 2.5]. The
proof of our next result is similar to that of the better known but weaker result for
ﬁnitely generated projectives which is a direct consequence of Nakayama’s Lemma.
Corollary 1.4. Let M ∈ M and suppose that F is a free module for which
there is an isomorphism F/mF
M/mM. Then there is an epimorphism L0F −→
Proof. The isomorphism F/mF
−→ M/mM lifts to a map F −→ M that
L0F −→ L0M
which has image N ⊆ M say. There is a commutative diagram
which shows that M = N + mM, so N = M.
Here is another example. Let S ⊆ m and let M = SM be the submodule of M
consisting of all sums of elements of the form sz for s ∈ S and z ∈ M. We say that
an R-module M is S-divisible if for every x ∈ M and s ∈ S, there exists y ∈ M
such that x = sy, i.e., M = SM. Since R is an integral domain, this is consistent
with Lam’s deﬁnition in chapter 1§3C of , see also corollary (3.17) .
Lemma 1.5. Let M ∈ M and let S ⊆ m be non-empty. If M is S-divisible then
it is trivial. In particular, injective objects in M are trivial.
Proof. For the ﬁrst statement, if M = SM then M ⊆ mM and so M = mM,
therefore M = 0.
Let M be injective in the category M . Then for each x ∈ M there is a
homomorphism R −→ M for which 1 → x. This extends to a homomorphism
L0R = R −→ M. For s ∈ S, there is a homomorphism L0R −→ L0R induced from
multiplication by s. By injectivity there is an extension to a diagram