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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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- 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.

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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.

I can't find a proof. Which is likely to mean the statement is false?

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- #2

Homework Helper

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I hope this works, since I'm a bit new to rings.

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Science Advisor

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Eg. consider the ideal generated by 4 over the integers.

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[This is a sign I should stop learning from various lecture notes, and turn to books. :uhh:]

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Science Advisor

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No. It is a sign you should think about what you're reading.

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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?

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- #9

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matt grime said:

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

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.

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.

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.

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.

An ideal generated by x^n is the smallest ideal that contains x^n. It is denoted as

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