PID Math Problem: Every Element Generating Ideal?

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In summary, a Principal Ideal Domain (PID) is a ring in which all ideals can be generated by a single element. However, this does not mean that every element in the ring can generate an ideal. Each element can generate a principle ideal, but a PID only guarantees that all ideals are principle ideals. Additionally, an ideal can be turned into a non-principle ideal by redefining its generating set, but it will still be considered a principle ideal.
  • #1
pivoxa15
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Homework Statement


If R is a PID then all its ideals can be generated by a single element. It dosen't imply that every element in R can generate an ideal does it?




The Attempt at a Solution


I have to agree that it dosen't but can't think of a proof.
 
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  • #2
Of course in any ring, any element can generate an ideal, the ideal generated by that element.
 
  • #3
You are right. Each element can always generate an ideal, namely a principle ideal. If R is a PID than it tells us that all the ideals in R are those that have been generated by a single element hence are all principle ideals.

However one can turn any principle ideal into a nonprinciple ideal. i.e. in Z. <3> is a Principle ideal. Instead of denoting it by <3> we denote this ideal by {0, 3} or {3, 6} and either of those two will be equivalent to <3>.
 
  • #4
It's still the same ideal, and is still a principal ideal, which is just an ideal that can be generated by a single element, regardless of whichever generating set you choose to focus on.
 

1. What is PID Math Problem: Every Element Generating Ideal?

PID Math Problem: Every Element Generating Ideal is a mathematical problem that involves finding a set of elements that generate an ideal in a given commutative ring. It is a fundamental concept in abstract algebra and has applications in various fields such as number theory, geometry, and physics.

2. How is PID Math Problem different from other mathematical problems?

PID Math Problem is different from other mathematical problems because it involves finding a set of elements that generate an ideal, rather than just solving for a specific equation or unknown variable. It also has a wide range of applications in various fields, making it a versatile and important concept in mathematics.

3. What is the significance of finding a PID in a commutative ring?

Finding a PID (Principal Ideal Domain) in a commutative ring is significant because it allows for the unique factorization of elements in that ring. This means that any element in the ring can be expressed as a product of prime elements in a unique way, similar to how integers can be factored into prime numbers.

4. What are some real-world applications of PID Math Problem?

PID Math Problem has various real-world applications, such as in cryptography, where it is used to create secure encryption algorithms. It also has applications in coding theory, where it is used to design efficient error-correcting codes. In physics, PID is used to describe the behavior of oscillating systems, such as springs and pendulums.

5. What are some strategies for solving PID Math Problem?

There are several strategies for solving PID Math Problem, including the Euclidean algorithm, which is used to find the greatest common divisor of two elements in a commutative ring. Other strategies include using the properties of prime elements and ideals to simplify the problem, as well as using properties of PID to reduce the problem to simpler cases.

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