Proceedings of the 44th Symposium on Ring Theory and ...

Remark 2. Let S 1 and S 2 be Serre subcategories <strong>of</strong> R-Mod.
(1) Since <strong>the</strong> zero module belongs to any Serre subcategory, one has S 1 ⊆ (S 1 , S 2 ) and
S 2 ⊆ (S 1 , S 2 ).
(2) It holds S 1 ⊇ S 2 if and only if (S 1 , S 2 ) = S 1 .
(3) It holds S 1 ⊆ S 2 if and only if (S 1 , S 2 ) = S 2 .
(4) A subcategory (S 1 , S 2 ) is closed under finite direct sums.
Example 3. We denote by S f.g. <strong>the</strong> subcategory consisting <strong>of</strong> finitely generated R-
modules and by S Artin <strong>the</strong> subcategory consisting <strong>of</strong> Artinian R-modules. If R is a complete
local ring, <strong>the</strong>n a subcategory (S f.g. , S Artin ) is known as <strong>the</strong> subcategory consisting
<strong>of</strong> Matlis reflexive R-modules. Therefore, (S f.g. , S Artin ) is a Serre subcategory <strong>of</strong> R-Mod.
The following example shows that a subcategory (S 1 , S 2 ) needs not be a Serre subcategory
for Serre subcategories S 1 and S 2 .
Example 4. We shall see that <strong>the</strong> subcategory (S Artin , S f.g. ) needs not be closed under
extensions.
Let R be a one dimensional Gorenstein local ring with a maximal ideal m. Then one
has a minimal injective resolution
0 → R → ⊕
E R (R/p) → E R (R/m) → 0
p ∈ Spec(R)
htp = 0
<strong>of</strong> R. (E R (M) denotes <strong>the</strong> injective hull <strong>of</strong> an R-module M.) We note that R and
E R (R/m) are in (S Artin , S f.g. ).
Now, we assume that a subcategory (S Artin , S f.g. ) is closed under extensions. Then
E R (R) = ⊕ htp=0 E R (R/p) is in (S Artin , S f.g. ). It follows from <strong>the</strong> definition <strong>of</strong> (S Artin , S f.g. )
that <strong>the</strong>re exists an Artinian R-submodule X <strong>of</strong> E R (R) such that E R (R)/X is a finitely
generated R-module.
If X = 0, <strong>the</strong>n E R (R) is a finitely generated injective R-module. It follows from <strong>the</strong>
Bass formula that one has dim R = depth R = inj dim E R (R) = 0. However, this equality
contradicts dim R = 1. On <strong>the</strong> o<strong>the</strong>r hand, if X ≠ 0, <strong>the</strong>n X is a non-zero Artinian
R-module. Therefore, one has Ass R (X) = {m}. Since X is an R-submodule <strong>of</strong> E R (R),
one has
Ass R (X) ⊆ Ass R (E R (R)) = {p ∈ Spec(R) | ht p = 0}.
This is contradiction as well.
3. The main result
In this section, we shall give a criterion for a subcategory (S 1 , S 2 ) to be a Serre subcategory
for Serre subcategories S 1 and S 2 .
First <strong>of</strong> all, it is easy to see that <strong>the</strong> following assertion holds.
Proposition 5. Let S 1 and S 2 be Serre subcategories <strong>of</strong> R-Mod.
(S 1 , S 2 ) is closed under submodules and quotient modules.
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Then a subcategory