---- > [!definition] Definition. ([[extension of an ideal]]) > Let $\varphi:A \to B$ be a [[ring homomorphism]]. The **extension** of an [[ideal]] $\mathfrak{a}$ of $A$ is the [[ideal]] $\mathfrak{a}^{e}:= \langle \varphi(\mathfrak{a}) \rangle$ of $B$ (i.e., the [[ideal]] of $B$ [[ideal generated by a subset|generated by]] the image of $\mathfrak{a}$ under $\varphi$). > Ideals of $B$ of the form $\mathfrak{a}^{e}$ are called **extended ideals**. > [!equivalence] > An ideal $\mathfrak{b}$ of $B$ is extended if and only if $\mathfrak{b}^{ce}=\mathfrak{b}$. > [!proof] Proof of equivalence. > $\to$. Suppose $\mathfrak{b}$ is extended: $\mathfrak{b}=\mathfrak{a}^{e}$ for some $\mathfrak{a}$. Then $\mathfrak{b}^{ce}=\mathfrak{a}^{ece}=\mathfrak{a}^{e}=\mathfrak{b}$ by the properties below. > $\leftarrow$. If $\mathfrak{b}^{ce}=\mathfrak{b}$, then $\mathfrak{b}=(\mathfrak{b}^{c})^{e}$ is obviously extended. ^proof ---- #### - $(\mathfrak{a}_{1} + \mathfrak{a}_{2})^{e}=\mathfrak{a}_{1}^{e}+\mathfrak{a_{}}_{2}^{e}$, since $\langle \varphi(\mathfrak{a}_{1} + \mathfrak{a}_{2}) \rangle = \langle \varphi(\mathfrak{a_{1}} ) + \varphi(\mathfrak{a}_{2}) \rangle = \langle \varphi(\mathfrak{a_{1}}) \rangle +\langle \varphi(\mathfrak{a_{2}}) \rangle .$ - $(\mathfrak{a}_{1}\mathfrak{a}_{2})^{e}=\mathfrak{a}_{1}^{e} \mathfrak{a}_{2}^{e}$, since $\langle \varphi(\mathfrak{a}_{1} \mathfrak{a}_{2}) \rangle = \langle \varphi(\mathfrak{a}_{1} ) \varphi(\mathfrak{a}_{2}) \rangle = \langle \varphi(\mathfrak{a}_{1}) \rangle \langle \varphi(\mathfrak{a}_{2}) \rangle . $ - **$\mathfrak{a} \subset \mathfrak{a}^{ec}$** since obviously for any $a \in \mathfrak{a}$, $\varphi(a) \in \langle \varphi(\mathfrak{a}) \rangle$, and hence $a \in \varphi ^{-1}(\langle \varphi(a) \rangle)$. - **$\mathfrak{b} \supset \mathfrak{b}^{ce}$** because $\varphi \varphi ^{-1}( \mathfrak{b} )\subset \mathfrak{b}$ by set theory and passing to $\mathfrak{b}^{ce}=\langle \varphi \varphi ^{-1}(\mathfrak{b}) \rangle$ preserves the inclusion ($\mathfrak{b}$ is an [[ideal]], and so is stable under taking linear combinations). - **$\mathfrak{b}^{c}=\mathfrak{b}^{cec}$.** $\supset$ follows immediately from the fact that $\mathfrak{b} \supset \mathfrak{b^{}}^{ce}$ and the general set theory fact $A \subset B$ $\implies$ $f^{-1}(A) \subset f^{-1}(B)$. $\subset$ follows from the general fact that $\mathfrak{a} \subset \mathfrak{a}^{ec}$, applied to $\mathfrak{a}=\mathfrak{b}^{c}$. - **$\mathfrak{a}^{e}=\mathfrak{a}^{ece}$.** $\subset$ follows immediately from the general fact that $\mathfrak{b} \supset \mathfrak{b^{}}^{ce}$, applied to $\mathfrak{b}=\mathfrak{a}^{e}$. $\supset$ follows from the fact that $\mathfrak{a} \subset \mathfrak{a}^{ec}$, hence $\varphi(\mathfrak{a}) \subset \varphi(\mathfrak{a}^{ec})$, hence $\langle \varphi(\mathfrak{a}) \rangle \subset \langle \varphi(\mathfrak{a}^{ec}) \rangle$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```