4 ANDREW BAKER

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 −→

M.

Proof. The isomorphism F/mF

∼

=

−→ M/mM lifts to a map F −→ M that

factors through

L0F −→ L0M

∼

=

M,

which has image N ⊆ M say. There is a commutative diagram

F

L0F

M

F/mF

∼

=

L0F/mL0F

∼

=

M/mM

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 [11], 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

0

L0R

s

L0R

M

4

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 −→

M.

Proof. The isomorphism F/mF

∼

=

−→ M/mM lifts to a map F −→ M that

factors through

L0F −→ L0M

∼

=

M,

which has image N ⊆ M say. There is a commutative diagram

F

L0F

M

F/mF

∼

=

L0F/mL0F

∼

=

M/mM

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 [11], 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

0

L0R

s

L0R

M

4