Associate elements - Revision history

Vipul at 01:46, 24 January 2009

2009-01-24T01:46:04Z

← Older revision		Revision as of 01:46, 24 January 2009
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	==Definition==		==Definition==

	Two elements in a [[commutative unital ring]] are said to be '''associate elements''' if ~~each~~ one ~~divides~~ the other. The relation of being associate elements is an equivalence relation.		Two elements in a [[commutative unital ring]] are said to be '''associate elements''' if they satisfy the following:

			* Each element is a divisor of the other element.
			* The [[principal ideal]] generated by one element equals the principal ideal generated by the other.

			The relation of being associate elements is an equivalence relation.

	{{proofat\|[[Associate element relation is an equivalence relation]]}}		{{proofat\|[[Associate element relation is an equivalence relation]]}}

Vipul at 21:25, 23 January 2009

2009-01-23T21:25:24Z

← Older revision		Revision as of 21:25, 23 January 2009
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	Two elements in a [[commutative unital ring]] are said to be '''associate elements''' if each one divides the other. The relation of being associate elements is an equivalence relation.		Two elements in a [[commutative unital ring]] are said to be '''associate elements''' if each one divides the other. The relation of being associate elements is an equivalence relation.

			{{proofat\|[[Associate element relation is an equivalence relation]]}}

	==Facts==		==Facts==

	In an integral domain, two elements are associate if and only if they are in the same orbit under the multiplicative action of the [[group of units]] of the ring (or in other words, there is an invertible element that multiplied with the first gives the second). ~~This is no longer true when the commutative unital ring has zero divisors.~~		In an integral domain, two elements are associate if and only if they are in the same orbit under the multiplicative action of the [[group of units]] of the ring (or in other words, there is an invertible element that multiplied with the first gives the second). Refer:

			* [[Elements in same orbit under multiplication by group of units are associate]]
			* [[Associate implies same orbit under multiplication by group of units in integral domain]]
			* [[Associate not implies same orbit under multiplication by group of units]]

Vipul: 1 revision

2008-05-12T16:18:44Z

1 revision

← Older revision	Revision as of 16:18, 12 May 2008
(No difference)

Vipul at 02:30, 9 August 2007

2007-08-09T02:30:08Z

New page

==Definition==

Two elements in a [[commutative unital ring]] are said to be '''associate elements''' if each one divides the other. The relation of being associate elements is an equivalence relation.

==Facts==

In an integral domain, two elements are associate if and only if they are in the same orbit under the multiplicative action of the [[group of units]] of the ring (or in other words, there is an invertible element that multiplied with the first gives the second). This is no longer true when the commutative unital ring has zero divisors.