Principal Ideals and associates

  • Thread starter Bleys
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In summary, the conversation discusses the relationship between principal ideals and associates in a K-field. The question is whether, if two non-constant elements f and h are associates, does it follow that their generated ideals are equal. The speaker attempts to prove this themselves and concludes that the equality is true because of the properties of units in K-fields. The other person clarifies a small error and the conversation ends with a summary of the proof.
  • #1
Bleys
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I'm wondering something about principal ideals which I'm using to prove something.
K-field, let f,h be non-constant in K[X]. If f and h are associates, does it follow (f) = (h) ?
I tried to just prove it myself but I'm not sure if it's correct.
f, h associates means f=ch for some unit c. Then (f) = {fg : g in K[X]} = {hcg : g in K[X]}. Now I want to put " = (h) " but I'm not sure if that's correct. I think it is, because g runs over all of K[X], and cK[X] = K[X], so {hcg : g in K[X]} = {hq : q in K[X]}. So is my statement true?
 
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  • #2
(f) = {fg : g in K[X]} = {hcg : g in K[X]} shows that any element in the ideal generated by f is in the ideal generated by g. Since c is a unit the converse is also true.
 
  • #3
lavinia said:
(f) = {fg : g in K[X]} = {hcg : g in K[X]} shows that any element in the ideal generated by f is in the ideal generated by g. Since c is a unit the converse is also true.
Did you mean h?

Also, is the equality {hcg : g in K[X]} = {hq : q in K[X]} incorrect?
 
  • #4
yes I meant h
 
  • #5
a = bu, where u is a unit. also from this you get b = au^-1. obviously a is in (b), which means that (a) is a subset of (b), and b is in (a), so (b) is a subset of (a). by double inclusion, they are equal.
 

1. What are principal ideals?

Principal ideals are sets of elements that can be generated by a single element, called the "generator" of the ideal. In other words, a principal ideal is a set of elements that can be obtained by multiplying a single element by all possible elements in the ring.

2. What is the significance of principal ideals in mathematics?

Principal ideals are important in understanding the structure of rings and modules. They help to simplify the study of more complex algebraic structures and can be used to prove theorems and solve equations in abstract algebra.

3. How are principal ideals related to ideal quotients?

Principal ideals are a special case of ideal quotients, which are sets of elements obtained by dividing an ideal by another ideal. In fact, every principal ideal is an ideal quotient of a ring by a principal ideal.

4. What are associates in the context of principal ideals?

Associates are elements in a ring that differ from each other only by a unit. In other words, two elements are associates if one can be obtained from the other by multiplying by a unit. In the context of principal ideals, associates are elements that generate the same principal ideal.

5. How do principal ideals and associates relate to prime and irreducible elements?

In a principal ideal domain (PID), every prime element is also irreducible, and every irreducible element generates a principal ideal. Additionally, in a PID, every non-zero non-unit element can be uniquely expressed as a product of irreducible elements, up to associates. This property is known as unique factorization.

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