Universal side divisor implies irreducible - Revision history

Vipul at 16:01, 5 February 2009

2009-02-05T16:01:22Z

← Older revision		Revision as of 16:01, 5 February 2009
Line 14:		Line 14:
	* [[Associate implies same orbit under multiplication by group of units in integral domain]]		* [[Associate implies same orbit under multiplication by group of units in integral domain]]
	* [[Associate not implies same orbit under multiplication by group of units]]		* [[Associate not implies same orbit under multiplication by group of units]]
	* [[Element of ~~smallest~~ norm among non-units in Euclidean ring is a universal side divisor]]		* [[Element of minimum norm among non-units in Euclidean ring is a universal side divisor]]
	* [[Euclidean ring that is not a field has a universal side divisor]]		* [[Euclidean ring that is not a field has a universal side divisor]]
	==Proof==		==Proof==

Vipul at 22:31, 31 January 2009

2009-01-31T22:31:07Z

← Older revision		Revision as of 22:31, 31 January 2009
Line 1:		Line 1:
	==Statement==		==Statement==

	In an [[~~integral domain~~]], any [[fact about::universal side divisor]] is an [[fact about::irreducible element]].		In an [[commutative unital ring]], any [[fact about::universal side divisor]] is an [[fact about::irreducible element]].

	==Related facts==		==Related facts==
Line 18:		Line 18:
	==Proof==		==Proof==

	'''Given''': ~~An integral domain~~ <math>R</math>, a universal side divisor <math>x \in R</math> such that <math>x = ab</math>.		'''Given''': A commutative unital ring <math>R</math>, a universal side divisor <math>x \in R</math> such that <math>x = ab</math>.

	'''To prove''': <math>x</math> is neither zero nor a unit, and either <math>a</math> is a unit or <math>b</math> is a unit.		'''To prove''': <math>x</math> is neither zero nor a unit, and either <math>a</math> is a unit or <math>b</math> is a unit.
Line 26:		Line 26:
	Since <math>x</math> is a universal side divisor, we obtain that either <math>x \| a</math> or there exists a unit <math>u</math> such that <math>x \| a - u</math>.		Since <math>x</math> is a universal side divisor, we obtain that either <math>x \| a</math> or there exists a unit <math>u</math> such that <math>x \| a - u</math>.

	# Case <math>x \| a</math>: We have <math>a ~~= xy</math> for some <math>y~~</math>, and ~~we get~~ <math>x ~~= xyb~~</math>, ~~yielding <math>x(1 - yb) = 0</math>. Since~~ <math>x ~~\ne 0~~</math> and <math>R</math> ~~is an integral domain~~, we ~~obtain <math>by = 1</math>, so <math>b</math> is a unit~~.		# Case <math>x \| a</math>: In this case we have <math>x \| a</math> and <math>a \| x</math>, so <math>x</math> and <math>a</math> are associates, and we are done.
	# Case there exists a unit <math>u</math> such that <math>x \| a - u</math>: We have <math>a - u = xy</math> for some <math>y</math>, yielding <math>u = a - xy = a - aby = a(1 - by)</math>. Since <math>u</math> is a unit, there exists <math>v</math> such that <math>uv = 1</math>, yielding <math>a(1-by)v = 1</math>, so <math>a</math> is a unit with inverse <math>(1-by)v</math>.		# Case there exists a unit <math>u</math> such that <math>x \| a - u</math>: We have <math>a - u = xy</math> for some <math>y</math>, yielding <math>u = a - xy = a - aby = a(1 - by)</math>. Since <math>u</math> is a unit, there exists <math>v</math> such that <math>uv = 1</math>, yielding <math>a(1-by)v = 1</math>, so <math>a</math> is a unit with inverse <math>(1-by)v</math>. Thus, <math>x</math> is associate with <math>b</math>, and we are done.

Vipul: New page: ==Statement== In an integral domain, any fact about::universal side divisor is an fact about::irreducible element. ==Related facts== ===Converse=== The converse is not true...

2009-01-31T17:54:39Z

New page: ==Statement== In an integral domain, any fact about::universal side divisor is an fact about::irreducible element. ==Related facts== ===Converse=== The converse is not true...

New page

==Statement==

In an [[integral domain]], any [[fact about::universal side divisor]] is an [[fact about::irreducible element]].

==Related facts==

===Converse===

The converse is not true, even in a [[Euclidean domain]]. {{further|[[Irreducible not implies universal side divisor]]}}

===Other related facts===

* [[Universal side divisor not implies prime]]
* [[Associate implies same orbit under multiplication by group of units in integral domain]]
* [[Associate not implies same orbit under multiplication by group of units]]
* [[Element of smallest norm among non-units in Euclidean ring is a universal side divisor]]
* [[Euclidean ring that is not a field has a universal side divisor]]
==Proof==

'''Given''': An integral domain <math>R</math>, a universal side divisor <math>x \in R</math> such that <math>x = ab</math>.

'''To prove''': <math>x</math> is neither zero nor a unit, and either <math>a</math> is a unit or <math>b</math> is a unit.

'''Proof''': The fact that <math>x</math> is neither zero nor a unit follows form the definition of universal side divisor, so it remains to show that if <math>x = ab</math>, then either <math>a</math> is a unit or <math>b</math> is a unit.

Since <math>x</math> is a universal side divisor, we obtain that either <math>x | a</math> or there exists a unit <math>u</math> such that <math>x | a - u</math>.

# Case <math>x | a</math>: We have <math>a = xy</math> for some <math>y</math>, and we get <math>x = xyb</math>, yielding <math>x(1 - yb) = 0</math>. Since <math>x \ne 0</math> and <math>R</math> is an integral domain, we obtain <math>by = 1</math>, so <math>b</math> is a unit.
# Case there exists a unit <math>u</math> such that <math>x | a - u</math>: We have <math>a - u = xy</math> for some <math>y</math>, yielding <math>u = a - xy = a - aby = a(1 - by)</math>. Since <math>u</math> is a unit, there exists <math>v</math> such that <math>uv = 1</math>, yielding <math>a(1-by)v = 1</math>, so <math>a</math> is a unit with inverse <math>(1-by)v</math>.