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The claim that if R is commutative and R_M is a field for all maximal idelas, then R is a field was wrong and so I deleted it.
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iravan
iravan
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If M is a maximal ideal of Z(R), the center of a ring with identity R, and R_M, the localization of R at M, is a commutative field, what can we say about R?

My guess is that we can's say that much because even for a commutative ring R, the only thing I can say is that for every x in M there exists some y in R\M such that xy = 0. Of course, it is easy to see that if R is commutative and R_M is a field for "all" maximal ideals of R, then R is a field.

Let me put a condition on R: suppose that R is von Neumann regular, that is for every r in R there exists some s in R such that r = rsr. If R_M is a commutative field for some maximal ideal M of Z(R), what can we say about R?

If M is a maximal ideal of Z(R), the center of a ring with identity R, and R_M, the localization of R at M, is a commutative field, what can we say about R?

My guess is that we can's say that much because even for a commutative ring R, the only thing I can say is that for every x in M there exists some y in R\M such that xy = 0. Of course, it is easy to see that if R is commutative and R_M is a field for "all" maximal ideals of R, then R is a field.

Let me put a condition on R: suppose that R is von Neumann regular, that is for every r in R there exists some s in R such that r = rsr. If R_M is a commutative field for some maximal ideal M of Z(R), what can we say about R?

If M is a maximal ideal of Z(R), the center of a ring with identity R, and R_M, the localization of R at M, is a commutative field, what can we say about R?

My guess is that we can's say that much because even for a commutative ring R, the only thing I can say is that for every x in M there exists some y in R\M such that xy = 0.

Let me put a condition on R: suppose that R is von Neumann regular, that is for every r in R there exists some s in R such that r = rsr. If R_M is a commutative field for some maximal ideal M of Z(R), what can we say about R?

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iravan
iravan
  • 59
  • 2

localization at central maximal ideals

If M is a maximal ideal of Z(R), the center of a ring with identity R, and R_M, the localization of R at M, is a commutative field, what can we say about R?

My guess is that we can's say that much because even for a commutative ring R, the only thing I can say is that for every x in M there exists some y in R\M such that xy = 0. Of course, it is easy to see that if R is commutative and R_M is a field for "all" maximal ideals of R, then R is a field.

Let me put a condition on R: suppose that R is von Neumann regular, that is for every r in R there exists some s in R such that r = rsr. If R_M is a commutative field for some maximal ideal M of Z(R), what can we say about R?