Section18.1Essential Extensions and Injective Hulls
Proposition18.1.
Let \(\theta: M \rightarrow N\) be a one-to-one map of \(R\)-modules. The following are equivalent:
For any map \(\alpha: N \rightarrow L, \alpha \circ \theta\) is one-to-one \(\Rightarrow \alpha\) is one-to-one. In particular, if \(M \subseteq N\text{,}\) then \(\left.\alpha\right|_{M}\) is one-to-one \(\Rightarrow \alpha\) is one-to-one.
Every nonzero submodule of \(N\) has a nonzero intersection with \(\theta(M)\text{.}\)
Every nonzero element of \(N\) has a nonzero multiple in \(\theta(M)\text{.}\)
Proof.
(1) \(\Rightarrow\) (2): If \(N^{\prime} \subseteq N\) is nonzero, then \(\alpha: N \rightarrow N / N^{\prime}\) is not one-to-one, hence \(\alpha \circ \theta: M \rightarrow\)\(N / N^{\prime}\) is not either, so \(M \cap N^{\prime} \neq 0\text{.}\)
\((2) \Rightarrow(3)\text{:}\) This is the special case of a cyclic submodule.
\((3) \Rightarrow(1)\text{:}\) Given a nonzero element in \(n \in \operatorname{ker}(\alpha), n\) has a nonzero multiple in \(\theta(M)\text{,}\) and any multiple of \(n\) is still in the kernel of \(\alpha\text{.}\)
Definition18.2.Essential Extension.
A map \(\theta\) that satisfies the equivalent conditions above is called an essential extension. We will often say that one module is an essential extension of another, meaning that the latter is a submodule, and the inclusion map is essential.
Roughly, one can think of \(A \subseteq B\) is an essential extension as being something like a dual property to \(A\) being a generating set for \(B\) (even though generating sets aren’t modules). If \(A \subseteq B\) is essential, then anything in \(B\) can get multiplied back into \(A\text{,}\) whereas if \(A\) is a generating set for \(B\text{,}\) anything in \(B\) can be obtained by multiplying from elements in \(A\) (and adding).
Example18.3.
If \(W\) is a multiplicative set consisting of nonzerodivisors on \(M\text{,}\) then \(W^{-1} M\) is an essential extension of \(M\text{.}\) In particular, the fraction field of a domain is an essential extension of it.
Example18.4.
If \(M\) is a (nonzero) \(I\)-torsion module, that is, every element of \(M\) is killed by a power of \(I\text{,}\) then \(\operatorname{ann}_{M}(I) \subseteq M\) is an essential extension.
Definition18.5.
In particular, if \((R, \mathfrak{m})\) is local and \(M\) is \(\mathfrak{m}\)-torsion, \(\operatorname{ann}_{M}(\mathfrak{m})\) is called the socle of \(M\)
Proposition18.6.
\(M\) is an essential extension of its socle.
Proof.
Given \(m \in M\text{,}\) we have that \(\operatorname{Ass}(R m) \subseteq \operatorname{Ass}(M){ }^{2}\text{.}\) 1
For a not-necessarily-finitely-generated module \(M, \operatorname{Ass}(M)\) is the set of primes \(\mathfrak{q}\) such that \(R / \mathfrak{q}\) embeds into \(M\text{.}\) Any nonzero module has an associated prime: this reduces to the finitely generated case by considering a nonzero finitely generated submodule.
Since \(M\) is \(I\)-torsion, \(\operatorname{Ass}(M) \subseteq V(I)\text{.}\) Let \(\mathfrak{p} \in \operatorname{Ass}(R m)\text{,}\) and \(x \in R m\) such that \(\operatorname{ann}(x)=\mathfrak{p}\text{.}\) Then \(x\) is a nonzero multiple of \(m\) that is killed by \(I\text{.}\)
Proposition18.7.
Let \(L \subseteq M \subseteq N\) be \(R\)-modules.
\(L \subseteq N\) is essential if and only if \(L \subseteq M\) and \(M \subseteq N\) are essential.
If \(M \subseteq N_{i} \subseteq N\) and \(N=\bigcup N_{i}\text{,}\) then \(M \subseteq N\) is essential if and only if each \(M \subseteq N_{i}\) is essential.
There is a unique \(N^{\prime}\) such that \(M \subseteq N^{\prime} \subseteq N\) that is maximal with respect to the property that \(M \subseteq N^{\prime}\) is essential.
Definition18.8.
A module \(N^{\prime}\) as in (3) above is called a maximal essential extension of \(M\) in \(N\text{.}\) If \(M \subseteq N\) is essential, and \(N\) has no proper essential extensions, then we say that \(N\) is a maximal essential extension of \(M\text{.}\) For a not-necessarily-finitely-generated module \(M, \operatorname{Ass}(M)\) is the set of primes \(\mathfrak{q}\) such that \(R / \mathfrak{q}\) embeds into \(M\text{.}\) Any nonzero module has an associated prime: this reduces to the finitely generated case by considering a nonzero finitely generated submodule.
Beware that there are two notions of maximal essential extension above: a relative version that takes place inside another module, and an absolute version.
Proposition18.9.
Let \(M\) be an \(R\)-module. The following are equivalent:
\(M\) is an injective module
every one-to-one map \(M \rightarrow N\) splits
\(M\) has no proper essential extensions
Proof.
(1) \(\Rightarrow(2)\text{:}\) The identity map \(M \rightarrow M\) extends to a map \(N \rightarrow M\) by the definition of injective.
\((2) \Rightarrow(3)\text{:}\) It suffices to note that \(M \subseteq M \oplus M^{\prime}\) is not essential: \(\left(0, m^{\prime}\right)\) has no nonzero multiple in (the image of) \(M\text{.}\)
\((3) \Rightarrow(1)\text{:}\) Embed \(M\) in an injective module, \(E\text{.}\) By Zorn’s lemma, there is a maximal submodule \(N\) of \(E\) such that \(M \cap N=0\text{.}\) Then, \(M \rightarrow E / N\) is essential, and by hypothesis, an isomorphism. Then, \(E=M+N=M \oplus N\text{.}\) Since \(M\) is a direct summand of an injective module, \(M\) is injective.
In particular, any maximal essential extension in the absolute sense is an injective module.
Proposition18.10.
Let \(M\) be an \(R\)-module. If \(E\) is an injective module with \(M \subseteq E\text{,}\) then the maximal essential extension of \(M\) in \(E\) is an injective module. All maximal essential extensions of \(M\) are isomorphic.
Proof.
Let \(E^{\prime}\) be a maximal essential extension of \(M\) in \(E\text{,}\) and \(Q\) be an essential extension of \(E^{\prime}\) (possibly not in \(E\) ). Since \(E\) is injective, the inclusion of \(E^{\prime} \subseteq E\) extends to a map from \(Q \rightarrow E\text{.}\) Since \(E^{\prime} \rightarrow E\) was one-to-one, and \(E^{\prime} \rightarrow Q\) essential, \(Q\) to \(E\) is one-to-one. By definition of \(E^{\prime}\text{,}\) we have \(E^{\prime}=Q\text{.}\) Thus, \(E^{\prime}\) is an absolute maximal essential extension of \(M\text{,}\) hence an injective module.
Let \(E^{\prime}\) and \(E^{\prime \prime}\) be two maximal essential extensions of \(M\text{.}\) The map from \(M \rightarrow E^{\prime \prime}\) extends to a map \(\varphi: E^{\prime} \rightarrow E^{\prime \prime}\text{.}\) Since \(E^{\prime}\) is an essential extension of \(M\) and \(M \rightarrow E^{\prime \prime}\) is one-to-one, \(\varphi: E^{\prime} \rightarrow E^{\prime \prime}\) is one-to-one. Since \(E^{\prime}\) is injective, \(E^{\prime \prime}=\varphi\left(E^{\prime}\right) \oplus C\) for some \(C\text{.}\) Since \(\varphi\left(E^{\prime}\right) \rightarrow E^{\prime \prime}\) is essential, \(C=0\text{.}\)
Definition18.11.
An injective hull or injective envelope of an \(R\)-module \(M\) is a maximal essential extension of \(M\text{.}\) By the previous proposition, this is well-defined up to isomorphism. We write \(E_{R}(M)\) for an injective hull of \(M\text{.}\)
Definition18.12.
A minimal injective resolution of an \(R\)-module \(M\) is an injective resolution \(M \rightarrow E^{\bullet}\) of \(M\) in which \(E^{0}=E_{R}(M)\text{,}\) and \(E^{i}=E_{R}\left(\operatorname{coker}\left(\partial_{E^{\bullet}}^{i}\right)\right)\) for each \(i>0\text{.}\)
