Vipul: New page: {{curing metaproperty satisfaction| property = Euclidean ring| metaproperty = quotient-closed property of commutative unital rings}} ==Statement== Suppose $R$ is a [[commutati...

2009-02-05T16:39:24Z

New page: {{curing metaproperty satisfaction| property = Euclidean ring| metaproperty = quotient-closed property of commutative unital rings}} ==Statement== Suppose <math>R</math> is a [[commutati...

New page

{{curing metaproperty satisfaction|
property = Euclidean ring|
metaproperty = quotient-closed property of commutative unital rings}}

==Statement==

Suppose <math>R</math> is a [[commutative unital ring]] that possesses a [[fact about::Euclidean norm]] <math>N</math>. Suppose <math>I</math> is an [[ideal]] inside <math>R</math>. Then, <math>R/I</math> is a Euclidean ring, with norm given by:

<math>\overline{N}(x + I) = \min \{ N(x + r) \mid r \in I \}</math>.

In other words, the norm of a coset is defined as the minimum of normso f all elements in the coset. (Note that this minimum is well-defined since it is the minimum over a nonempty subset of a well-ordered set.

==Related facts==

* [[Minimum over principal ideal of Euclidean norm is a smaller multiplicatively monotone Euclidean norm]]
* [[Euclideanness is localization-closed]]

==Proof==

'''Given''': A commutative unital ring <math>R</math> with Euclidean norm <math>N</math>. An ideal <math>I</math> of <math>R</math>. <math>\overline{N}</math> is defined on <math>R/I</math> by:

<math>\overline{N}(x + I) = \min \{ N(x + r) \mid r \in I \}</math>.

'''To prove''': <math>\overline{N}</math> is a Euclidean norm on <math>R/I</math>.

'''Proof''': Suppose <math>a + I, b + I</math> are two elements of <math>R/I</math> with <math>b + I \ne I</math>. Now, there exists <math>c \in a + I, d \in b + I</math> with <math>N(c) = \overline{N}(a)</math> and <math>N(d) = \overline{N}(b)</math>. By the Euclidean division in <math>R</math> we have:

<math>c = dq + r</math>

where <math>r = 0</math> or <math>N(r) < N(d)</math>. Going modulo <math>I</math>, we get:

<math>c + I = (d + I)(q + I) + (r + I)</math>

which can be rewritten as:

<math>a + I = (b + I)(q + I) + (r + I)</math>

where <math>r + I = I</math> or <math>N(r) < N(d)</math>. Note that <math>\overline{N}(r + I) \le N(r)</math> by definition, and <math>N(d) = \overline{N}(b + I)</math> by the choice of <math>d</math>. Thus, we have <math>r + I = I</math> or <math>\overline{N}(r + I) < \overline{N}(b + I)</math>, which is precisely the condition for Euclidean division in <math>R/I</math>.

Euclideanness is quotient-closed - Revision history

Vipul: New page: {{curing metaproperty satisfaction| property = Euclidean ring| metaproperty = quotient-closed property of commutative unital rings}} ==Statement== Suppose R is a [[commutati...

Vipul: New page: {{curing metaproperty satisfaction| property = Euclidean ring| metaproperty = quotient-closed property of commutative unital rings}} ==Statement== Suppose $R$ is a [[commutati...