Show ring ideal is not principal ideal

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In summary, the conversation is about proving that the ideal (3, x^3 - x^2 + 2x -1) in the ring of polynomials over the integers is not a principal ideal. The attempt at a solution involved showing that if it were a principal ideal, it would have to equal the entire ring, but since 1 is not an element of the ideal, it cannot be a principal ideal.
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AcidRainLiTE
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Homework Statement


Show that the ideal
[tex] (3, x^3 - x^2 + 2x -1) \text{ in } \mathbb{Z}[x] [/tex]
is not principal. (The parentheses mean 'the ideal generated by the elements enclosed in parentheses')

2. The attempt at a solution
I came up with a solution (see attachment), it is just rather convoluted. I feel like I am missing a more informative proof. I would like to take away as much as I can from this problem, so I am wondering if someone else sees a more insightful way to prove it. A brief summary of my solution is (see attachment for details):

(1) If it (call it I) is a principal ideal, then the fact that it contains 3 implies
[tex] I = a \mathbb{Z}[x] \text{ for some } a \in \mathbb{Z} [/tex]

(2) But the fact that it contains [tex]x^3 - x^2 + 2x -1[/tex] implies a = 1 so
[tex] I = \mathbb{Z}[x] [/tex]

(3) But [itex] 1 \notin I [/itex] and hence [itex] I \ne \mathbb{Z}[x] [/itex]. Contradiction. So I is not a principal ideal.
 
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What is a show ring ideal?

A show ring ideal is a set of qualities or characteristics that judges in a dog show or other animal competition look for when evaluating animals. These qualities may include physical appearance, behavior, and training.

What is a principal ideal?

A principal ideal is an ideal in abstract algebra that is generated by a single element. In other words, it is the set of all elements that can be obtained by multiplying the generator by any element in the underlying ring.

How are show ring ideals and principal ideals related?

They are not necessarily related. The term "show ring ideal is not principal ideal" means that the qualities or characteristics that are sought after in a show ring are not necessarily the same as those that are considered ideal in abstract algebra.

Why is this concept important in the scientific community?

This concept is important because it highlights the differences between subjective and objective ideals. Show ring ideals are based on personal preferences and opinions, while principal ideals are based on mathematical principles and properties.

Is there any practical application of this concept?

Yes, understanding the concept of show ring ideal vs principal ideal can help scientists and researchers recognize the difference between subjective and objective standards in their work. It can also help them better understand how different fields of study may have different ideals and goals.

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