My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
November 16th, 2010, 09:27 AM   #1
Newbie
 
Joined: Nov 2010

Posts: 10
Thanks: 0

Ring homomorphisms and kernel

Hi!

In a book about number theory (Number fields - Daniel A.Marcus), he talk about the inertial degree. There is two steps here I don`t fully understand.

He let K and L be numberfields where K? L and R=A?K og S=A?L (Where A is the set of algebraic integer in C)
P and Q are non zero prime ideals, where P is in R and Q in S. Futhermore we know that Q lies over P.

question 1:
this is the point I wonder about : R? S => R -> S/Q
i.e. The containment of R in S induces a ringhomomorphism "R -> S/Q". Why is that?

Of course we know that the following is required for a ring homomorphism:
i) f(a+b) = f(a) + f(b) and ii) f(ab) = f(a)f(b)

As I understand, the inclusion "?" serves as our "f" over. We have that:
i) f(a +R b) = f(a) +s f(b)
ii) f(a *R b) = f(a) *s f(b)

question 2:
From the above the book concludes that the kernel must be R?Q
Why is that?

I would be really greatfull if somebody could explain to me more in detail about these questions.
Thank you so much for your time and effort.

Thomas
norway9k is offline  
 
November 16th, 2010, 11:05 PM   #2
Senior Member
 
Joined: Aug 2010

Posts: 195
Thanks: 5

Re: Ring homomorphisms and kernel

For your first question, note that since is an ideal of , there is a homomorphism which takes an element to its congruence class modulo . Furthermore, we have the inclusion map which is also a homomorphism. Composing these maps gives us a rather natural homomorphism .

For the second question, note that has kernel exactly , and since the inclusion map is injective (and hence has kernel only the zero element), under the composition of maps, an element in will go to zero in if and only if the inclusion map sends an element of into , but since this map is just inclusion, these are exactly the elements in .
Turgul is offline  
November 17th, 2010, 12:28 AM   #3
Newbie
 
Joined: Nov 2010

Posts: 10
Thanks: 0

Re: Ring homomorphisms and kernel

Ah, ok I see. Thank you
norway9k is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
homomorphisms, kernel, ring



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
ring homomorphisms tiger4 Abstract Algebra 0 April 20th, 2012 10:21 PM
Galois Theory questions: Homomorphisms wattsup Abstract Algebra 0 November 2nd, 2011 05:45 PM
True/False about Homomorphisms and Zero Divisors jkong05 Abstract Algebra 1 May 5th, 2009 03:41 AM
commutative diagram, module homomorphisms fermatprime371 Abstract Algebra 0 December 8th, 2008 07:44 PM
homomorphisms mathstresser Abstract Algebra 0 November 18th, 2007 02:00 PM





Copyright © 2018 My Math Forum. All rights reserved.