Vipul: 2 revisions

2008-05-12T16:33:56Z

2 revisions

← Older revision	Revision as of 16:33, 12 May 2008
(No difference)

Vipul at 01:06, 8 January 2008

2008-01-08T01:06:01Z

← Older revision		Revision as of 01:06, 8 January 2008
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	# <math>R</math> is a [[Jacobson ring]]		# <math>R</math> is a [[Jacobson ring]]
	# If <math>P</math> is a [[prime ideal]] of <math>R</math> and if <math>S := R/P</math> contains an element <math>b \ne 0</math> such that <math>S[b^{-1}]</math> is a [[field]], then <math>S</math> is a field		# If <math>P</math> is a [[prime ideal]] of <math>R</math> and if <math>S := R/P</math> contains an element <math>b \ne 0</math> such that <math>S[b^{-1}]</math> is a [[field]], then <math>S</math> is a field

			==Proof==

			===1 implies 2===

			As in the hypotheses, let <math>P</math> be a [[prime ideal]] in <math>R</math>. Then <math>S := R/P</math> is an [[integral domain]], and hence 0 is a prime ideal in this. Further, <math>S</math> is also a [[Jacobson ring]] since [[Jacobson is quotient-closed]], and thus the zero ideal in <math>S</math> is an [[intersection of maximal ideals]].

			Now, the primes of <math>S[b^{-1}]</math> correspond to the primes of <math>S</math> that do not contain <math>b</math>. Since <math>S[b{-1}]</math> is a [[field]], <math>b</math> is contained in ''every'' nonzero prime ideal of <math>S</math>. Thus if <math>S</math> were ''not'' a [[field]], zero would not be a maximal ideal, and hence <math>b</math> would be contained in every [[maximal ideal]], contradicting the fact that the intersection of all maximal ideals is zero. Thus <math>S[b^{-1}]</math> is a field.

			===2 implies 1===

			{{fillin}}

Vipul at 00:57, 8 January 2008

2008-01-08T00:57:49Z

New page

==Statement==

Let <math>R</math> be a [[commutative unital ring]]. The following are equivalent:

# <math>R</math> is a [[Jacobson ring]]
# If <math>P</math> is a [[prime ideal]] of <math>R</math> and if <math>S := R/P</math> contains an element <math>b \ne 0</math> such that <math>S[b^{-1}]</math> is a [[field]], then <math>S</math> is a field

Rabinowitch's trick - Revision history

Vipul: 2 revisions

Vipul at 01:06, 8 January 2008

Vipul at 00:57, 8 January 2008