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
February 10th, 2018, 01:56 AM   #1
Member
 
Joined: Sep 2011

Posts: 98
Thanks: 1

Ring

Hi, I have attached the question and the solutions to part a and b of this question. Would like someone to verify if I have done anything wrong especially with part b. Greatly appreciate it! Thanks.
Attached Images
File Type: jpg q2.jpg (20.3 KB, 3 views)
File Type: jpg Webp.net-resizeimage.jpg (68.1 KB, 9 views)
File Type: jpg 11.jpg (73.6 KB, 12 views)
Alexis87 is offline  
 
February 10th, 2018, 03:16 AM   #2
Global Moderator
 
Joined: Dec 2006

Posts: 19,702
Thanks: 1804

If you can't achieve clearer images, I would suggest typing your work.
skipjack is online now  
February 10th, 2018, 07:43 AM   #3
Senior Member
 
Joined: Aug 2017
From: United Kingdom

Posts: 264
Thanks: 79

Math Focus: Algebraic Number Theory, Arithmetic Geometry
I'd say this is fine except I'm not sure what you mean by "S is a commutative ring for all $[a], [b] \in Z_{10}$." Surely you just mean it's a commutative ring?

It's worth saying that if a ring $R$ is commutative, then the commutativity of any subring $T$ follows immediately. Indeed, for any $a, b \in T$, note that $a$ and $b$ are in $R$ so they commute. This sort of argument is why we only need to check a few things to show something is a subring - properties like associativity of +, associativity of x, distributivity, follow immediately by the same sort of argument.
cjem is online now  
February 10th, 2018, 09:06 AM   #4
Banned Camp
 
Joined: Apr 2017
From: durban

Posts: 22
Thanks: 0

Math Focus: Algebra
The subring test is a theorem that states that for any ring R, a subset of R is a subring if it is closed under multiplication and subtraction, and contains the multiplicative identity of R.
Ola Lawson is offline  
February 10th, 2018, 09:08 AM   #5
Banned Camp
 
Joined: Apr 2017
From: durban

Posts: 22
Thanks: 0

Math Focus: Algebra
I think you can finish it from here. but looking at the first expression the working is right with Z10
Ola Lawson is offline  
February 10th, 2018, 09:24 AM   #6
Senior Member
 
Joined: Aug 2017
From: United Kingdom

Posts: 264
Thanks: 79

Math Focus: Algebraic Number Theory, Arithmetic Geometry
Quote:
Originally Posted by Ola Lawson View Post
The subring test is a theorem that states that for any ring R, a subset of R is a subring if it is closed under multiplication and subtraction, and contains the multiplicative identity of R.
The subset $S \subseteq Z_{10}$ in the question does not contain the multiplicative identity of $Z_{10}$, so it wouldn't actually be a subring under the usual definition (the one you're working with). However, the question still says to prove it's a subring, so they're probably using a different definition.

Last edited by cjem; February 10th, 2018 at 09:27 AM.
cjem is online now  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
ring



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Ring limes5 Abstract Algebra 1 June 29th, 2015 12:24 AM
Ring Mathew Abstract Algebra 5 August 29th, 2010 08:53 PM
ring tinynerdi Abstract Algebra 4 April 4th, 2010 10:17 PM
Ring and pseudo-ring cgouttebroze Abstract Algebra 5 August 14th, 2008 12:04 PM
ring Frazier001 Abstract Algebra 1 December 6th, 2007 01:21 PM





Copyright © 2018 My Math Forum. All rights reserved.