## Ideals

Last update: 01 February 2012

## Ideals

Let $R$ be a ring.

• A fractional ideal is
• An invertible ideal is
• A prime ideal is
• A primary ideal is
• A primitive ideal is
• A divisorial ideal is
• The prime spectrum of $R$ is
• The inverse of an ideal $I$ is $I - 1 = x∈𝕂 xI⊆R ,$ where $𝕂$ is the field of fractions of $R$.
• The Zariski topology on $\mathrm{Spec}\left(R\right)$ is
• The class group of $R$ is the group $Cl(R)= Inv(R) Frac(R) .$
• The class group of $R$ is the group $Cl(R)= Div(R) Prin(R) .$
• The Picard group of $R$ is the group $Pic(R)= Inv(R) Prin(R) .$

1. Let $R$ be an integral domain. If $R$ is a UFD or $R$ is quasilocal then $\mathrm{Pic}\left(R\right)=0$.
2. Let $R$ be an algebraic number ring. If $R$ is a PID then $\mathrm{Cl}\left(R\right)=0$.

Examples.

• $\mathrm{Cl}\left(ℤ\left[i\right]\right)=0.$
• $\mathrm{Cl}\left(ℤ\left[\sqrt{-5}\right]\right)=\frac{ℤ}{2ℤ}.$

(Nagata's theorem.) This relates the class group $\mathrm{Cl}\left(R\right)$ of $R$ and the class group $\mathrm{Cl}\left({R}_{S}\right)$ of a localization of $R$.

Example. Let $D$ be an integral domain and let $𝕂$ be the field of fractions of $D$. Let $R=D [{x}^{2},{x}^{3}] .$ Then $\mathrm{Pic}\left(𝕂 [{x}^{2},{x}^{3}] \right)=𝕂.$ See Lam's book.

## Notes and References

Where are these from?

References?