X in ideal if x^n is in ideal?

  • Thread starter pivoxa15
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In summary, the statement that if x^n is in an ideal, then x must also be in the ideal is false. This is because the ideal generated by x^n in a ring is not necessarily guaranteed to contain x as well. This can be seen through examples, such as the ideal <2^2> in the ring of natural numbers, which contains no element 2. Therefore, the statement is not universally true and is only applicable to prime ideals.
  • #1
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Homework Statement


In a ring with multiplicative identity, If x^n is in an ideal then is x also in the ideal? with n a natural number.

The Attempt at a Solution


I can't find a proof. Which is likely to mean the statement is false?
 
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  • #2
Let x^n be in the ideal I of a ring R. It can be represented as x^n = x^(n-1) x. Now, either both x^(n-1) and x are in the ideal (in that case the statement if proved), or one of them is in the ideal. If x is in the ideal, the proof is finished. If x^(n-1) is in the ideal, then you can apply the same argument again.

I hope this works, since I'm a bit new to rings.
 
  • #3
The statement that a*b in an ideal implies a and/or b in the ideal is only true for prime ideals. Not any ideal. The statement is false.
 
  • #4
Eg. consider the ideal generated by 4 over the integers.
 
  • #5
[This is a sign I should stop learning from various lecture notes, and turn to books. :uhh:]
 
  • #6
No. It is a sign you should think about what you're reading.
 
  • #7
pivoxa15 said:
I can't find a proof. Which is likely to mean the statement is false?


Just think about an example for a second. Like the simplest ring there is, the integers. Why didn't you do some examples to see if it was true?
 
  • #8
Right, I should have done that. So the ideal generated by <x^2> in a ring certainly wouldn't have x in it. So the statement is false.
 
  • #9
It is perfectly possible for the ideal generated by x^2 to have x in it. It is just not *certain* to have x in it.
 
  • #10
matt grime said:
It is perfectly possible for the ideal generated by x^2 to have x in it. It is just not *certain* to have x in it.

Good point. An example might be the ideal <2^2> in N which contains no element 2.
 

1. What is an ideal?

An ideal is a subset of a ring that satisfies certain properties. In simple terms, it is a collection of elements that can be multiplied by any element in the ring and still remain within the ideal.

2. How is an ideal related to x^n?

If x^n is an element of an ideal, it means that the ideal contains all possible powers of x up to the nth power. This is because an ideal must contain all products of elements within the ideal, and x^n can be written as a product of x multiplied by itself n times.

3. Why is x^n in ideal significant?

X^n being in an ideal is significant because it allows for the ideal to generate all other powers of x through multiplication. This is useful in many mathematical applications, such as in polynomial rings and algebraic geometry.

4. Can x^n be in ideal if x is not in ideal?

Yes, x^n can still be in an ideal even if x is not in the ideal. This is because the ideal only needs to contain all possible powers of x, not necessarily x itself.

5. What is an ideal generated by x^n?

An ideal generated by x^n is the smallest ideal that contains x^n. It is denoted as and is equal to the set of all multiples of x^n, including 0.

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