Turn on thread page Beta
    • Thread Starter
    Offline

    1
    ReputationRep:
    I am trying to understand the concept of Ideals in ring theory, and so far I think I get it but would like some confirmation.

     (a) = Ra \{ra: r \epsilon R \} is the ideal generated by  a

    (By the way I know there exist left and right ideals but I do not need to know this as I am working with commutative rings).

    I have:

     (a_1,a_2) = \{r_1a_1+r_2a_2 : r_1, r_2 \epsilon R \} etc for ideals with more elements.

    There is a line in my notes which is:

    "Sometimes you don't need two elements:  (15,21) \mathbb{Z} = (3) "

    I understand that this works, but what I do not get is how you can tell whether two ideals are equal.

    Is there a way to tell that two ideals are equal or is it just inspection.

    I also think the way I have written the line in my notes is wrong. I mean I understand that it means:

    Both  (15,21) and  (3) are ideals for the ring  \mathbb{Z} and that  (15,21) = (3) , but is there a way in which to write this without saying it in words.

    Thanks in advance.
    • PS Helper
    Offline

    14
    PS Helper
    Let's use the (15,21) and (3) example.

    Now, the hcf of 15 and 21 is 3, and so there exist integers a and b such that 15a + 21b = 3; so in particular, 3 \in (15, 21). But then notice that 3r = 15ar+21br \in (15,21), so any multiple of 3 is in the ideal, and so (3) \subseteq (15,21).

    Also, if x \in (15,21) then x=15a+21b=3(5a+7b) for some a,b, and so x \in (3). In other words, (15,21) \subseteq (3). But by the previous result, this means that (3)=(15,21).

    In general, we can define "highest common factors" in commutative rings. We say that d is a highest common factor of a and b if d|a,\ d|b and if d'|a and d'|b then d'|d. These can be useful for checking if two rings are equal, but not always (since we can't always write d=ax+by).
    Offline

    18
    ReputationRep:
    (Original post by nuodai)
    In general, we can define "highest common factors" in commutative rings. We say that d is a highest common factor of a and b if d|a,\ d|b and if d'|a and d'|b then d'|d. These can be useful for checking if two rings are equal, but not always (since we can't always write d=ax+by).
    You can define them, but I think it's worth pointing out that they do not have to exist.
    • Thread Starter
    Offline

    1
    ReputationRep:
    Thanks
 
 
 

University open days

  1. University of Cambridge
    Christ's College Undergraduate
    Wed, 26 Sep '18
  2. Norwich University of the Arts
    Undergraduate Open Days Undergraduate
    Fri, 28 Sep '18
  3. Edge Hill University
    Faculty of Health and Social Care Undergraduate
    Sat, 29 Sep '18
Poll
Which accompaniment is best?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.