## Localization

Last update: 29 March 2014

## Localization

Let $R$ be a commutative ring.

• A prime ideal is an ideal $𝔭$ of $R$ such that $R/𝔭$ is an integral domain.
• A maximal ideal is an ideal $𝔪$ of $R$ such that $R/𝔪$ is a field.
• A local ring is a ring with a unique maximal ideal.
• A multiplicative subset is a subset $S\subseteq R$ which is closed under multiplication.

Exercise: An ideal $𝔭$ of $R$ is prime if and only if $R\setminus 𝔭$ is a multiplicative subset of $R$.

Let $R$ be a commutative ring, let $S$ be a subset of $R$ and let $𝔭$ be a prime ideal of $R$.

• The multiplicative closure of $S$ is the smallest multiplicative subset $\stackrel{_}{S}$ such that $\stackrel{_}{S}\supseteq S$.
• The ring of fractions defined by $S$ is the ring $R\left[{S}^{-1}\right]=\left\{\frac{a}{s}|a\in R,\phantom{\rule{.5em}{0ex}}s\in \stackrel{_}{S}\right\}$ $with a1 s1 = a2 s2 if there exists z ∈ S_ such that z ( a1 s2 - s1 a2 ) = 0 ,$ with operations given by $a1 s1 + a2 s2 = a1 s2 + s1 a2 s1 s2 and a1 s1 ⋅ a2 s2 = a1 a2 s1 s2 ,$ and with the map $R ↪ R [S-1] given by r↦r1 .$
• The localization of $R$ at $𝔭$ is ${R}_{𝔭}=R\left[{S}^{-1}\right]$ where $S=R\setminus 𝔭.$

Let $R$ be a ring, let $𝔭$ be a prime ideal of $R$ and let ${R}_{𝔭}=\left\{\frac{a}{d}|a,d\in R,d\notin 𝔭\right\}$ be the localization of $R$ at $𝔭$.

1. The functor $R𝔭⊗-: {R-modules} → {R𝔭-modules}, M ↦ M𝔭 where M𝔭 = R𝔭⊗RM,$ is exact.
2. The map ${proper ideals of R𝔭} ↔ {proper ideals ofR contained in 𝔭} I ↦ I∩R J𝔭 ↔ J$ is a bijection which takes prime ideals to prime ideals and ${𝔭}_{𝔭}$ is the unique maximal ideal of ${R}_{𝔭}.$

 Proof. Let $𝔮$ be a prime ideal of $R$ contained in $𝔭$. Then $R/𝔮$ is an integral domain and ${R}_{𝔭}/{𝔮}_{𝔭}$ is an integral domain??? a. ?? b. ?? This proof needs to be filled in. $\square$

## Notes and References

Where are these from?

References?