]> Commutative Algebra - Extension and Contraction

## Extension and Contraction

Let $f:R\to S$ be a ring homomorphism. Let $I◃R,J◃S$. The extension ${I}^{e}$ of $I$ (with respect to $f$) is $⟨f\left(I\right)⟩$. Then contraction ${J}^{c}$ of $J$ (with respect to $f$) is ${f}^{-1}\left(J\right)$+++i, which is an ideal in $R$+.

Note that if $J$ is prime than so is ${J}^{c}$, though the same is not always true for extensions. For example, take the identity map $f:ℤ\to ℚ$. Then for any prime $p$, $pℤ$ is prime in $ℤ$ but $\left(pℤ{\right)}^{e}=ℚ$ which is not prime in $ℚ$.

In general there is no simple relationship between the prime ideals of $R$ and $S$. For example consider the identity map $f:ℤ\to ℤ\left[i\right]$ (see notes on <a href="../numberfield"> number fields</a>).

Other properties of extension and contraction:

1. $I\subset {I}^{ec},J\supset {J}^{ce}$

2. ${J}^{c}={J}^{cec},I={I}^{ece}$

3. Let $𝒞$ be the set of contracted ideals in $R$ and $ℰ$ be the set of extended ideals in $S$. Then $𝒞=\left\{K◃R\mid {K}^{ec}=K\right\}$, $ℰ=\left\{L◃S\mid {L}^{ce}=L\right\}$, and $K↦{K}^{e}$ for all $K\in 𝒞$ defines a bijection $𝒞\to ℰ$ whose inverse is $L↦{L}^{c}$ for all $L\in ℰ$.

Let ${I}_{1},{I}_{2}◃R$ and ${J}_{1},{J}_{2}◃S$. Then

1. $\left({I}_{1}+{I}_{2}{\right)}^{e}={I}_{1}^{e}+{I}_{2}^{e}$, $\left({J}_{1}+{J}_{2}{\right)}^{c}\supset {J}_{1}^{c}+{J}_{2}^{c}$

2. ${I}_{1}\cap {I}_{2}{\right)}^{e}\subset {I}_{1}^{e}\cap {I}_{2}^{e}$, $\left({J}_{1}\cap {J}_{2}{\right)}^{c}={J}_{1}^{c}\cap {J}_{2}^{c}$

3. $\left({I}_{1}{I}_{2}{\right)}^{e}={I}_{1}^{e}{I}_{2}^{e}$, $\left({J}_{1}{J}_{2}{\right)}^{c}\supset {J}_{1}^{c}{J}_{2}^{c}$

4. $\left({I}_{1}:{I}_{2}{\right)}^{e}\subset \left({I}_{1}^{e}:{I}_{2}^{e}\right)$, $\left({J}_{1}:{J}_{2}{\right)}^{c}\subset \left({J}_{1}^{c}:{J}_{2}^{c}\right)$

5. $\left(\sqrt{{I}_{1}}{\right)}^{e}\subset \sqrt{{I}_{1}^{e}}$, $\left(\sqrt{{J}_{1}}{\right)}^{c}=\sqrt{{J}_{1}^{c}}$

The set $ℰ$ is closed under sum and product while the set $𝒞$ is closed under intersection, forming ideal quotients and taking radicals.