Problem with PID definition

Patrick Reany

13 July 2024

A PID in ring theory is a principal ideal domain (to be explained in a moment). I think that the definition of it, as it's commonly presented, has a bit of a problem. So, I asked ChatGPT-4o to help me work it out. Let's see what happened.

Patrick:
I'm a bit confused about the exact meaning of a principal ideal domain (PID) in ring theory: In a PID, every ideal is principal. Maybe you can help me figure it out.

ChatGPT-4o:
A Principal Ideal Domain (PID) is a specific type of ring in ring theory, a branch of abstract algebra. Here’s a detailed explanation to help clarify the concept:

Definition of a PID

A Principal Ideal Domain (PID) is an integral domain R in which every ideal I is a principal ideal. An ideal I in a ring R is called principal if there exists an element a \in R such that I can be generated by a. In other words,
I = (a) = { ra | r \in R }.

Key Properties of a PID

1. Integral Domain: A PID must be an integral domain, which means it is a commutative ring with unity (1 ≠ 0) and has no zero divisors.

2. Principal Ideals: Every ideal in a PID can be expressed as (a) for some a \in R.

Examples of PIDs

1. Integers \mathbb{Z} : The ring of integers is a PID. Every ideal in \mathbb{Z} is of the form (n) = { kn | k \in \mathbb{Z} } for some integer n.

2. Polynomial Ring k[x]: If k is a field, then the ring of polynomials k[x] is a PID. Every ideal in k[x] is generated by a single polynomial.

Non-Examples

1. Ring of Continuous Functions: The ring of all continuous functions from \mathbb{R} to \mathbb{R}, denoted by C(\mathbb{R}), is not a PID because there exist ideals that are not principal.

2. \mathbb{Z}[x]: The ring of polynomials with integer coefficients is not a PID. For example, the ideal (2, x) cannot be generated by a single element.

Understanding Ideals in a PID

To further understand what it means for every ideal to be principal in a PID, consider the following:

- In the ring of integers \mathbb{Z}, any ideal is of the form (d) where d is the greatest common divisor (gcd) of the integers in the ideal.
- In k[x], any ideal can be expressed as (f(x)) where f(x) is a polynomial. This is because polynomials over a field can always be factored into irreducibles, ensuring that any combination of them can be represented by a single polynomial.

Why is this Important?

The concept of PIDs is crucial because they provide a simpler structure to study within ring theory. In a PID, many properties and theorems are easier to prove due to the simpler nature of the ideals involved. For example:

- Factorization: In a PID, unique factorization can be guaranteed.
- Modules: The structure theory for modules over a PID is well-understood, akin to the classification of finitely generated abelian groups.

Summary

A Principal Ideal Domain (PID) is an integral domain where every ideal is principal, meaning it can be generated by a single element. This definition simplifies the study of ideals and their properties within the ring, making PIDs a fundamental concept in algebra.

If you have any further questions or need examples to illustrate these points, feel free to ask!

Patrick:
Okay, here we go. There are proofs in ring theory about quotient rings, say by ideal I of ring R, in which a hypothetical ideal J is place between I and R, and the goal of which is to show that either J = I or J = R. But in the latter case, since J is an ideal, then R would have to be an ideal of itself. The problem is therefore if R must also be a principal ideal of itself? Case a) if R is unital, then R = (1) and we're okay. Case b) R is not unital, then does that force us to conclude that R is a principal ideal of a ring larger that itself? Or, should we just define a principal ideal domain as a ring in which every proper ideal is principal?

ChatGPT-4o:
Let's clarify the points you raised and then address the issues with the definition of a principal ideal domain (PID).

Quotient Rings and Hypothetical Ideal J

When considering a quotient ring R/I and a hypothetical ideal J such that I \subseteq J \subseteq R, there are cases to consider:
1. J = I
2. J = R

If J is an ideal, the second case implies that R is an ideal of itself. Let's explore this further:

Ring R as an Ideal of Itself

In general, a ring R is always an ideal of itself. This follows from the definition of an ideal:
- For any r, s \in R , we have r - s \in R and for any r \in R and a \in R , we have ar \in R and ra \in R.
- Therefore, R satisfies the conditions of being an ideal of itself.

Principal Ideal of Itself

When we talk about principal ideals, we typically focus on proper ideals. A proper ideal is an ideal that is strictly contained within the ring, i.e., I \subsetneq R .

Case a) Unital Ring

If R is unital (i.e., has a multiplicative identity 1), then R can be generated by 1, i.e., R = (1). Here, R is trivially a principal ideal of itself because it can be generated by a single element, 1.

Case b) Non-Unital Ring

If R is not unital, the situation is different:
- In non-unital rings, we don't have a multiplicative identity.
- Defining a principal ideal in a non-unital ring requires a different approach since there is no single element generating the entire ring.

Principal Ideal Domain (PID) Definition

Given these considerations, it is useful to refine the definition of a PID:

Standard Definition:

A Principal Ideal Domain (PID) is an integral domain in which every ideal is principal.

Refined Definition:

A Principal Ideal Domain (PID) is an integral domain in which every proper ideal is principal.

Clarification:

1. Integral Domain: R is an integral domain, meaning it is a commutative ring with no zero divisors.
2. Principal Proper Ideals: Every proper ideal I (i.e., I \subsetneq R ) is principal, meaning I = (a) for some a \in R .

This refined definition aligns with the usual context in ring theory and abstract algebra, where we are often interested in the structure of proper ideals within the ring. By focusing on proper ideals, we avoid the complications arising from considering the ring as an ideal of itself, especially in non-unital rings.

Conclusion

To address your concern directly:
- In a unital ring, R = (1) is trivially a principal ideal of itself.
- In a non-unital ring, the notion of the ring being a principal ideal of itself doesn't apply in the same way, and proper ideals should be the focus.

Therefore, defining a Principal Ideal Domain as a ring in which every proper ideal is principal provides a clearer and more consistent framework, accommodating both unital and non-unital rings.

If you have further questions or specific examples you'd like to explore, please let me know!

My Comments: Basically, it's my fault as an oversight. Since PID R is a domain, it must have a unity and therefore R can be represented as (1), hence, R is itself principal. But, then again, it's not that simple, because I have found definitions of integral domains that do not mention the ring having a unity element. Thus, there is good reason to define what you mean by an integral domain before defining what you mean by a PID.

ChatGPT-4o defined an integral domain as having a unity, so perhaps it was confused about what I thought the actual problem is, though, when it condescended to define an integral domain as a commutative ring with no zero-divisors, it adapted my recommendations to this alternative system.

Question: Is it worth considering PID rings if the rings have no unity element? Perhaps so, and in which case we would probably only be interested in proper ideals.