Maximal ideal question, + invertibility

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In summary, the conversation discusses the concept of maximal ideals and prime ideals in commutative rings with and without unity. The result that in a commutative ring with unity, an ideal is maximal iff it is prime is mentioned. Two questions are posed, one regarding an example of a maximal ideal that is not prime in a finite commutative ring without unity, and another about a subring of matrices with a different unity. The conversation also touches on the idea of Artinian rings and non-unital rings. Finally, an example is given for a finite commutative ring without unity where an ideal is maximal but not prime. The conversation ends with the acknowledgement that non-unital rings are difficult to work with.
  • #1
sihag
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I'm unable to think of an example of a maximal ideal that is not prime in a finite commutative ring without unity.

Of course the context is, the result stating that in a commutative ring with unity, an ideal is maximal iff it is prime.

Somebody help me out with this ?

Another question is for the ring of 2 x 2 matrices over reals (lets call it M), its subring of matrices with all entries equal (lets call it R) is a ring with a different unity. (wrt usual matrix operations of addition and multiplication)

the unity is 1/2 in each entry.
Of course all non-zero elements of R are invertible, the inverse being (1/4)x (where x is the 'equal entry' for the element of R).
My question is why does this happen, though such matrices belonging to R are all singular.

My own reasoning is that since the determinant is 0 for all matrices belonging to R, we are really not defining anything by the determinant calculation. Also the adjoint of any element of R doesn't belong to R in this case, since the entries will not be equal for adjoint (negatives for the (2,1)th and (1,2)th entry).
But, I'm really not convinced by my own reasons. I know it's tantalizing. Does it have to with the necessary and sufficient criterion relating the determinant with the fact of invertibility of the matrix?

Help me out with this too, someone?

Thank you.
 
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  • #2
sihag said:
I'm unable to think of an example of a maximal ideal that is not prime in a finite commutative ring without unity.

Of course the context is, the result stating that in a commutative ring with unity, an ideal is maximal iff it is prime.

Somebody help me out with this ?
Well, obviously, you have to find where the proof for the unital case fails. Have you tried reviewing the proof with a fine-toothed comb to find where the failure happens? Here, I think the only problem is (hint in white) that if you have an ascending chain of proper ideals, you aren't guaranteed their union is proper. (In particular, this means you're going to have to look at rngs that aren't Noetherian) (highlight to read)


... My question is why does this happen, though such matrices belonging to R are all singular. ...
Well, I'm not really sure what you mean by "why". But my best guess is that you should look at algebraic properties -- can you think of any unusual equations that the identity of the subrng must satisfy? In particular (additional hint colored white) you might be interested in the idempotents of your rng (Highlight to read, but only after you've reflected on what I've already said!)
 
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  • #3
sihag said:
Of course the context is, the result stating that in a commutative ring with unity, an ideal is maximal iff it is prime.
This is not true. For example, in Z[x], a commutative ring with unity, <x> is prime but not maximal. But your statement is true for Artinian rings (and for PIDs).
 
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  • #4
Hurkyl said:
Well, obviously, you have to find where the proof for the unital case fails.
Hrm... I think I was answering a question different than the one you meant to ask with this section.
 
  • #5
Yeah - it looks like you were considering the question of whether or not a proper ideal sits inside a maximal ideal.

As for the original question, my gut reaction was to look at non-Artinian rings, and examine an infinite descending chain of ideals (since ideals are in particular non-unital rings). This actually did produce an example: in Z, we have the infinite descending chain 2Z > 4Z > 8Z > ..., and 4Z is a maximal but not prime ideal of the rng 2Z.

But then I realized that the OP was asking for an example in a finite commutative ring without unity. At first I thought this would be impossible to find, since finite rings are trivially Artinian, but I think I did manage to construct an example. Let S=Z/8Z and let R=2S (the ideal of S generated by 2). R is a finite commutative ring without unity (it has no nonzero idempotents). Now if we let I={0,4}, then this is an ideal of R that is maximal but not prime.

In conclusion: non-unital rings suck.
 
  • #6
mistake, i meant in a finite commutative ring with unity, an ideal is maximal iff it is prime.
yes it should work, if we use {0,2,4,6} addition and multiplication modulo 8
{0,4} is maximal but not prime. (2*6=4,under mult. modulo 8, but neither 2 nor 6 belong to the ideal). got it.


Hurkyl said:
Well, I'm not really sure what you mean by "why".

What i meant was, it feels counter-intuitive. I've got to think about it more.
 

Related to Maximal ideal question, + invertibility

1. What is a maximal ideal?

A maximal ideal is a proper subset of a ring (a mathematical structure) that is closed under addition and multiplication and contains no other proper ideals that are properly contained within it.

2. How is a maximal ideal different from a prime ideal?

A prime ideal is a proper subset of a ring that is closed under addition and multiplication and has the additional property that if the product of two elements is in the ideal, then at least one of the elements is also in the ideal. A maximal ideal is always a prime ideal, but not all prime ideals are maximal ideals.

3. What is the importance of maximal ideals in algebra?

Maximal ideals play a crucial role in the study of rings and fields in abstract algebra. They are used to define quotient rings, which are important in understanding the structure of rings. They also have applications in commutative algebra, algebraic geometry, and number theory.

4. How is invertibility related to maximal ideals?

A ring element is invertible if and only if it is not contained in any maximal ideal. This is known as the "maximal ideal theorem" and is a fundamental concept in the study of rings. Invertibility is important because it allows us to define a multiplicative inverse for a given element, which is crucial in solving equations and manipulating algebraic expressions.

5. Can a maximal ideal be non-invertible?

Yes, a maximal ideal can contain non-invertible elements. In fact, a maximal ideal that contains a non-invertible element is called a singular ideal. These types of ideals are important in the study of commutative rings and have applications in algebraic geometry.

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