Fourth Isomorphism Theorem

== Theorem ==
Let $\phi: R \to S$ be a ring homomorphism.
Let $K = \map \ker \phi$ be the kernel of $\phi$.
Let $\mathbb K$ be the set of all subrings of $R$ which contain $K$ as a subset.
Let $\mathbb S$ be the set of all subrings of $\Img \phi$.
Let $\phi^\to: \powerset R \to \powerset S$ be the direct image mapping of $\phi$.
Then its restriction $\phi^\to: \mathbb K \to \mathbb S$ is a bijection.
Also:
:$(1): \quad \phi^\to$ and its inverse both preserve subsets.
:$(2): \quad \phi^\to$ and its inverse both preserve ideals:
:::: If $J$ is an ideal of $R$, then $\map {\phi^\to} J$ is an ideal of $S$
:::: If $J'$ is an ideal of $S$, then $\map {\paren {\phi^\to}^{-1} } {J'}$ is an ideal of $R$
== Proof ==
=== Proof of Preservation of Subsets ===
From Subset Maps to Subset, we have:
:$(a) \quad \forall X, X' \in \mathbb K: X \subseteq X' \implies \map {\phi^\to} X \subseteq \map {\phi^\to} {X'}$
:$(b) \quad \forall Y, Y' \in \mathbb S: Y \subseteq Y' \implies \map {\paren {\phi^\to}^{-1} } Y \subseteq \map {\paren {\phi^\to}^{-1} } {Y'}$
So $\phi^\to$ and its inverse both preserve subsets, and $(1)$ has been demonstrated to hold.
=== Proof that Inverse Image is a Subring ===
Let $U \in \mathbb S$, that is, let $U$ be a subring of $\Img \phi$.
From Preimage of Subring under Ring Homomorphism is Subring, we have that $\map {\paren {\phi^\to}^{-1} } U$ is a subring of $R$ such that $K \subseteq R$ and so:
:$\map {\paren {\phi^\to}^{-1} } U \in \mathbb K$
=== Proof that Image is a Subring ===
Let $V \in \mathbb K$, that is, a subring of $R$ containing $K$.
From Ring Homomorphism Preserves Subrings, we have that $\map {\phi^\to} V$ is a subring of $S$ and so:
:$\map {\phi^\to} V \in \mathbb S$
=== Proof that $f$ is a Bijection ===
By Subset of Codomain is Superset of Image of Preimage, we have that:
:$U = \map {\phi^\to} {\map {\paren {\phi^\to}^{-1} } U}$
We also have from Subset of Domain is Subset of Preimage of Image that:
:$V \subseteq \map {\paren {\phi^\to}^{-1} } {\map {\phi^\to} V}$
Now let $r \in \map {\paren {\phi^\to}^{-1} } {\map {\phi^\to} V}$.
Thus $\map \phi r \in \map {\phi^\to} V$ and so:
:$\exists v \in V: \map \phi r = \map \phi v$
So $\map \phi {r + \paren {-v} } = 0_S$ and so $r - v \in K$ by definition of kernel.
So:
:$\exists k \in K: r + k = v$
But by assumption, $K \subseteq V$ as $V \in \mathbb K$.
So $k \in V$ and so it follows that $r \in V$ as well, by the fact that $V$ is subring and so closed for $+$.
So:
:$\map {\paren {\phi^\to}^{-1} } {\map {\phi^\to} V} \subseteq V$
Putting that together with $V \subseteq \map {\paren {\phi^\to}^{-1} } {\map {\phi^\to} V}$ and we see:
:$V = \map {\paren {\phi^\to}^{-1} } {\map {\phi^\to} V}$
So we have:
:$U = \map {\phi^\to} {\map {\paren {\phi^\to}^{-1} } U}$
:$V = \map {\paren {\phi^\to}^{-1} } {\map {\phi^\to} V}$
That is:
:$\phi^\to \circ \paren {\phi^\to}^{-1} = I_\mathbb S$
:$\paren {\phi^\to}^{-1} \circ \phi^\to = I_\mathbb K$
where $I_\mathbb S$ and $I_\mathbb K$ are the identity mappings of $\mathbb S$ and $\mathbb K$ respectively.
By Composite of Bijection with Inverse is Identity Mapping, it follows that $\phi^\to: \mathbb K \to \mathbb S$ is a bijection.
=== Proof of Preservation of Ideals ===
Let $V \in \mathbb K$ be an ideal of $R$.
Let $U = \map {\phi^\to} V$.
Then from Ring Epimorphism Preserves Ideals, $U$ is an ideal of $S$.
Similarly, let $U = \map {\phi^\to} V$ be an ideal of $S$.
Then by Preimage of Ideal under Ring Homomorphism is Ideal, $V = \map {\paren {\phi^\to}^{-1} } U$ is an ideal of $R$ such that $K \subseteq V$.
Hence ideals are preserved in both directions.
== Also see ==
* Isomorphism Theorems
== Sources ==
* : $\S 2.2$: Homomorphisms: Theorem $2.12$
Category:Quotient Rings
Category:Named Theorems
Category:Isomorphisms (Abstract Algebra)
Category:Isomorphism Theorems