PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (37)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: High
[parent] overring (Definition)

Let $ R$ be a commutative ring having regular elements and let $ T$ be the total ring of fractions of $ R$. Then $ R \subseteq T$. Every subring of $ T$ containing $ R$ is an overring of $ R$.

Example. Let $ p$ be a rational prime number. The $ p$-integral rational numbers are the quotients of two integers such that the divisor is not divisible by $ p$. The set of all $ p$-integral rationals is an overring of $ \mathbb{Z}$.



"overring" is owned by pahio.
(view preamble)

View style:

See Also: a condition of algebraic extension


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: rationals, divisible, integers, quotients, rational prime, subring, total ring of fractions, regular elements, commutative ring
There are 2 references to this entry.

This is version 9 of overring, born on 2004-05-21, modified 2008-05-10.
Object id is 5867, canonical name is Overring.
Accessed 1817 times total.

Classification:
AMS MSC: 13B30 (Commutative rings and algebras :: Ring extensions and related topics :: Quotients and localization)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

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