PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: High Entry average rating: No information on entry rating
fraction field (Definition)

Given an integral domain $R$ , the fraction field of $R$ is the localization $S^{-1} R$ of $R$ with respect to the multiplicative set $S = R \setminus \{0\}$ . It is always a field.




"fraction field" is owned by djao.
(view preamble | get metadata)

View style:

See Also: localization, rational function

Other names:  field of fractions, quotient field
Log in to rate this entry.
(view current ratings)

Cross-references: field, multiplicative set, localization, integral domain
There are 67 references to this entry.

This is version 3 of fraction field, born on 2001-10-19, modified 2002-03-04.
Object id is 394, canonical name is FractionField.
Accessed 13152 times total.

Classification:
AMS MSC13B30 (Commutative rings and algebras :: Ring extensions and related topics :: Quotients and localization)

Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy
isn't this... by drini on 2001-11-08 23:59:11
...also known as quotiens field?

(well.. is it the quotient field I learned in my 1st ring theory course.. am not sure .. never seen it like you show)
 f
G -----> H G
p \ /_ ----- ~ f(G) 
 \ / f ker f 
 G/ker f 
[ reply | up ]
non-commutative version by antizeus on 2001-10-19 23:01:39
Much like localization, there's a lovely theory devoted to non-commutative versions of this, which deserve treatment. Maybe I'll write something later.
[ reply | up ]

Interact
post | correct | update request | add derivation | add example | add (any)