Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 29 March 2014


Let R be a commutative ring.

Exercise: An ideal 𝔭 of R is prime if and only if R𝔭 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.

Let R be a ring, let 𝔭 be a prime ideal of R and let R𝔭 = { ad | a,dR, d𝔭 } be the localization of R at 𝔭.

  1. The functor R𝔭-: {R-modules} {R𝔭-modules}, M M𝔭 where M𝔭 = R𝔭RM, is exact.
  2. The map {proper ideals of R𝔭} {proper ideals ofR contained in 𝔭} I IR J𝔭 J is a bijection which takes prime ideals to prime ideals and 𝔭𝔭 is the unique maximal ideal of R𝔭.

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.

Notes and References

Where are these from?



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