Abstract Algebra Question: Maximal Ideals

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Homework Statement



a) Show that there is exactly one maximal ideal in Z_8 and in Z_9.

b) Show that Z_10 and Z_15 have more than one maximal ideal.


Homework Equations



I know a maximal ideal is one that is not contained within any other ideal (except for the ring itself)

By Theorem, we know that In a commutative ring R with identity, every maximal ideal is prime.


The Attempt at a Solution



For a) I was thinking I would just show that all of the classes were subsets of the other classes. i.e. [8/0] is contained in [4] is contained in [2], and [6] is contained in [3] and [2], [9] is contained in [3]. Does that make sense? But I couldn't figure out what to do with [5] and [7]. It seems to me like BOTH of those are maximal ideals, but I'm supposed to prove that there's only one. Also, not quite sure how to formalize this into a proof.

I'm pretty confident on what to do for b). I just have to show that there's more than one, right? And both Z_7 and Z_9 should be ideals in Z_10 and Z_15, aren't they?

Thanks!

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
Dick
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Instead of just writing [5], which I'm assuming is the ideal generated 5 in, say Z_8, it might make it clearer if you wrote out the elements contained in the ideal. Like, [4] in Z_8 is {0,4}, right? What's [5]?
 
  • #3
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Instead of just writing [5], which I'm assuming is the ideal generated 5 in, say Z_8, it might make it clearer if you wrote out the elements contained in the ideal. Like, [4] in Z_8 is {0,4}, right? What's [5]?

Oh, dang. Equivalence classes aren't ideals. Wow. Not sure what I was thinking.

All right. So it turns out that I actually have NO idea what I'm doing. Guess it's back to the drawing board.
 
  • #4
Dick
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Oh, dang. Equivalence classes aren't ideals. Wow. Not sure what I was thinking.

All right. So it turns out that I actually have NO idea what I'm doing. Guess it's back to the drawing board.

Oh, you meant [5] to be an equivalence class? No, an ideal of Z_8 is a subset of Z_8 that's also subring with another property. Better check the definition.
 
  • #5
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Oh, you meant [5] to be an equivalence class? No, an ideal of Z_8 is a subset of Z_8 that's also subring with another property. Better check the definition.

Yeah. I know the definition of ideal. I've just been doing abstract algebra for the last few hours, and I think my brain may have gone a little soft and mushy.

Thanks for the willingness to help!
 
  • #6
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Yeah. I know the definition of ideal. I've just been doing abstract algebra for the last few hours, and I think my brain may have gone a little soft and mushy.

Thanks for the willingness to help!

Some things to consider:

1) What is the prime factorization of 8? Of 9?

2) What do you know about the order of a subring with respect to its "parent ring"

3) Giving the theorem you stated, what are the eligible orders for a maximal ideal in Z_8? What about Z_9?
 

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