Hey,

Need some help figuring out what a maximal ideal is: the question is to list all maximal ideals in Z_8 and Z_10. The maximal ideal is similiar to the normal subgroup, but I would think there would only be one per ring.

Thanks for the help--

4/16/2006 1:09:56 PM

I think you're right. As I recall the maximal ideal is the biggest one not equal to the ring itself, lets see Wolfram has a nice defintion and some comments relevant to your question

Quote :

"A maximal ideal of a ring R is an ideal I, not equal to R, such that there are no ideals "in between" I and R. In other words, if J is an ideal which contains I as a subset, then either J=I or J=R. For example, nZ is a maximal ideal of Z iff n is prime, where Z is the ring of integers. Only in a local ring is there just one maximal ideal. For instance, in the integers, a=<p> is a maximal ideal whenever p is prime. A maximal ideal m is always a prime ideal, and the quotient ring A/m is always a field. In general, not all prime ideals are maximal. "

It would seem that Z_1 , Z_2, Z_3, Z_5, Z_7 are all maximal ideals of Z_8. Of course, I could be wrong... if so please tell me why.

4/16/2006 7:21:06 PM

I'll give you a couple of hints. First, no, there is not just one maximal ideal per ring; there can be arbitrarily many distinct maximal ideals.

Second, if R is a ring and I is a maximal ideal, the quotient ring R/I is a field. That limits the possible cardinalities of R/I quite a bit (in the finite case, the cardinality must be the power of a prime).

4/16/2006 7:21:58 PM

mathman, I didn't have time to respond to you yesterday when I made my previous post.

Anyway, first, your notation is a bit confusing. It's very nonstandard to refer to, for example, Z_7 as an ideal (maximal or not) of Z_8.

(In Z_8, I'll use 0,1,2,...,7 to denote the respective equivalence classes)

The only sensible way to interpret it, of course, is as the ideal generated by 7, generally denoted by (7).

However, 7 is a unit: 7*7=1. If an ideal contains a unit, the ideal is the entire ring itself, and cannot be maximal.

4/17/2006 5:37:07 PM

from how I understand it, which is a pretty weak understanding, the order of an ideal has to devide the order of the ring, which I believe has already been said in different words. But with this, it's also apparent that in Z_8, <7>, which is relatively prime (and thus generates Z_8), cannot be a maximal ideal, since as Cabbage said, it's the entire ring.

I think if the ideals have the same order, say, |<2>| = |<6>| in Z_8, then you just say that only <2> would be maximal, but I'd really like somebody else's opinion on this as well.

Note: I'm responding more to make myself think through this again rather than for virga's sake.

4/18/2006 12:40:31 AM

Quote :

"I think if the ideals have the same order, say, |<2>| = |<6>| in Z_8, then you just say that only <2> would be maximal, but I'd really like somebody else's opinion on this as well."

There can be two different maximal ideals that also have the same order.

For example, say you have the product ring Z_3 x Z_3. There are two distinct maximal ideals, both with three elements: The ideal generated by (1,0), and the one generated by (0,1).

In Z_8, (2) and (6) are both maximal ideals having order 4. They are not distinct, however, since 2*3=6, and so (2) = (6): They are the same ideal.

4/18/2006 4:03:23 AM

yeah.. that they weren't distinct was more what I meant. Thanks for clarifying.

4/18/2006 8:28:45 AM

Actually, <6> is not an ideal of Z_8 because six does not divide 8 ---

When finding the maximal ideals I've learned that you list out all the of the divisors of the n, and then find the orders of those elements, and see what is contained in what -- like drawing the lattice. What is at the top are the maximal ideals.

omg 1600th post

[Edited on April 18, 2006 at 10:46 AM. Reason : 1600]

4/18/2006 10:37:05 AM

Quote :

"Actually, <6> is not an ideal of Z_8 because six does not divide 8"

I have no idea what you mean. What's your definition of the symbol "<6>" then?

If you're using it the same way I'm using "(6)", then it's defined as the ideal generated by 6 (the smallest ideal containing 6), so of course it's an ideal, simply by definition.

(0), (1), (2), ..., (7), are all (not necessarily maximal) ideals of Z_8, simply by definition, whether the generating element divides 8 (in the integers) or not.

4/18/2006 2:39:54 PM

Quote :

"When finding the maximal ideals I've learned that you list out all the of the divisors of the n, and then find the orders of those elements, and see what is contained in what -- like drawing the lattice. What is at the top are the maximal ideals."

I should have commented on this as well, because you are correct about this (but I think it's clear it only works for Z_n, and not for arbitrary rings (though there may be some generalization you could make to other rings)).

The divisors of 8 (in the integers) are 1,2,4, and 8. Considering the ideals generated by these elements individually, only (2) is maximal.

As a set, (2)={0,2,4,6}. 4 does not generate this entire ideal, but 6 does. (6) must contain the elements:

6*1 = 6
6*2 = 4
6*3 = 2
6*4 = 0

So (6) contains everything (2) does (and vice versa), they're the same ideal.

There's nothing special about the element 2 that requires you to write this ideal only as (2); it's just as valid to write this ideal as (6), as well.

4/18/2006 2:53:26 PM

Just to make one more thing clear (hopefully it's already clear, but I'm not sure):

If I was grading answers to the problem, "Find all maximal ideals of Z_8" and saw the following answers:

A. (2) is the only maximal ideal

B. (6) is the only maximal ideal

C. (0,2,4,6) is the only maximal ideal

D. (2) and (6) are the only maximal ideals

(Of course, I would expect there to be some jusitification for the answers, but let's only look at the final answers for now).

A, B, and C are all entirely correct, they simply expressed the same ideal in different manners. There are a few other natural ways to express it as well, such as (2,6) (the ideal generated by the set {2,6}). (2) and (6) seem to me the most elegant ways to describe the ideal (because of their conciseness, and I see no particular reason to prefer one over the other), but all are correct.

D would bother me. It implies to me that the student believes (2) and (6) are distinct ideals, when in fact they are the same.

4/18/2006 6:02:22 PM

Hey Cabbage, thanks for your comments. It's been a while since I've gotten my hands dirty with the modular arithmatic examples. I see my error, wolfram was talking about maximal ideals of the integers, but we are interested in maximal ideal(s) of Z mod 8.

To summarize,

(1)= Z_8
(2)=(6)={2,4,6,0} (generating with 2)
(3)=(5)=(7)={3,6,1,4,7,2,5} (generating with 3)
(4)={4,0}

So Z_8 has additive subgroups of order 2 and 4 (which is 2^2=prime^2). These are also subrings since they are closed under multiplication mod 8 (both left and right). Then to make them an ideal we need that they are closed under (left/right) multiplication by the whole ring Z_8,

{0,4}"times"{0,1,2,3,4,5,6,7}={0,4,8,12,16,20,24,28}={0,4} mod 8.

the same goes for (2) it also closes back on (2) under multiplication by the whole Ring Z_8. So (2) and (4) are not just additive subgroups, they are ideals. By explicit computation we can see they are the only "proper" ideals of Z_8, hence (2)=(6) is the maximal ideal of Z_8.

Of course, my presentation here is one rooted in abject ignorance to the many nice labor saving theorems that you have to play with. Memories of the Euler-phi function dance in my head, but the dance is blurred by the passing years I'm afraid...

4/18/2006 6:09:24 PM

No problem, mathman, glad to help. Everything's perfect, except for one nitpick (which looks more like a typo or omission rather than a real mistake):

Quote :

"(3)=(5)=(7)={3,6,1,4,7,2,5} (generating with 3)"

You forgot 0! (3)={3,6,1,4,7,2,5,0}

So we have (1)=(3)=(5)=(7)={3,6,1,4,7,2,5,0}={0,1,2,3,4,5,6,7}=Z_8.

Summarizing, there are exactly four ideals in Z_8:

A. (1) (which can also be written (3) or (5) or (7) or other (less efficient) ways. This ideal is the entire ring, hence it is not maximal.

B. (4) = {0,4}

C. (2) (or (6), or other less efficient ways of writing it)

D. Don't forget the trivial ideal generated by 0: (0) = {0}

A and D are both trivial, the entire ring and the zero ideal, respectively.

B is a proper ideal, but not maximal.

C is the only maximal ideal.

4/18/2006 6:21:27 PM

cabbage got it

[Edited on April 18, 2006 at 6:27 PM. Reason : .]

4/18/2006 6:26:38 PM

Quote :

"You forgot 0!"

but I had 1.

sorry. very sorry. well not that much.

4/18/2006 6:34:36 PM