Ideals

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $$I^{-1}= {x\in 𝕂|xI\subseteq R} ,$$ 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 $$\mathrm{Cl}\left(R\right)=\frac{\mathrm{Inv}\left(R\right)}{\mathrm{Frac}\left(R\right)}.$$
• The class group of $R$ is the group $$\mathrm{Cl}\left(R\right)=\frac{\mathrm{Div}\left(R\right)}{\mathrm{Prin}\left(R\right)}.$$
• The Picard group of $R$ is the group $$\mathrm{Pic}\left(R\right)=\frac{\mathrm{Inv}\left(R\right)}{\mathrm{Prin}\left(R\right)}.$$

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.

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

(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

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