Question

Show that in a PID, all non-zero prime ideals are maximal (see Exercise 7.38 ). Recall that for a complex number \(\alpha=a+b i\), with \(a, b \in \mathbb{R},\) the norm of \(\alpha\) was defined as \(N(\alpha)=\alpha \bar{\alpha}=a^{2}+b^{2}\) (see Example 7.5). There are other measures of the "size" of a complex number that are useful. The absolute value of \(\alpha\) is defined as \(|\alpha|:=\sqrt{N(\alpha)}=\sqrt{a^{2}+b^{2}}\). The max norm of \(\alpha\) is defined as \(M(\alpha):=\max \\{|a|,|b|\\}\)

Short answer

Explain your answer. Answer: Yes, in a Principal Ideal Domain (PID), all non-zero prime ideals are maximal. This is because for a given non-zero prime ideal in a PID, there are no ideals properly contained between this prime ideal and the entire ring. The proof involves considering an ideal containing the prime ideal, showing that the generator of that ideal divides the generator of the prime ideal, and finally concluding that either the ideal is equal to the prime ideal or to the entire ring, satisfying the condition of a maximal ideal.

Step-by-step solution

Definition of a PID

A Principal Ideal Domain (PID) is an integral domain in which every ideal is generated by a single element. In other words, any ideal I in a PID can be written as I = (a), where a is the generator of I and (a) denotes the ideal generated by a.

Definition of Prime Ideal

An ideal P is called a prime ideal in a ring R if the following conditions hold: 1. P is a proper ideal, i.e., P ≠ R 2. If ab ∈ P for elements a, b ∈ R, then either a ∈ P or b ∈ P.

Definition of Maximal Ideal

An ideal M is called a maximal ideal in a ring R if the following conditions hold: 1. M is a proper ideal, i.e., M ≠ R. 2. There are no ideals properly contained between M and R. In other words, if J is an ideal such that M ⊆ J ⊆ R, then either J = M or J = R.

Let P be a non-zero prime ideal in a PID R

Let R be a PID and suppose P is a non-zero prime ideal in R. We will aim to prove that P is a maximal ideal. Note that since R is a PID, we can write P = (p) for some non-zero element p ∈ R.

Consider an ideal J containing P

Let J be an ideal of R containing P such that P ⊆ J ⊆ R. Since R is a PID, we can write J = (j) for some element j ∈ R.

Show that the generator j of J divides p

Since P = (p) ⊆ J = (j), it must hold that p ∈ J. It means that p = jk for some k ∈ R. Therefore, the generator j of J divides p.

Show that either J = P or J = R

From Step 3, we have p = jk. We know that p ∈ P, so by the definition of prime ideal, either j ∈ P or k ∈ P. If j ∈ P (note that this implies P = J), then J = (j) = P and there is no proper containment. If j ∉ P, then k ∈ P, which means k = pr for some r ∈ R. In this case, we can express 1 = k/p = jr, which implies that j and p are associates in the PID, and we have J = (j) = (p) = P. Thus, either J = P or J = R.

Conclusion

Given that there are no ideals properly contained between P and R, we have proven that all non-zero prime ideals are maximal in a PID. In addition, we have discussed the definitions and properties of the norm, the absolute value, and the max norm for complex numbers, which are alternative "size" measures of complex numbers.

Key concepts

Principal Ideal Domain (PID)

A Principal Ideal Domain (PID) is a fundamental concept in algebra, especially within ring theory. Essentially, a PID is a type of ring with some very nice properties that make it easier to work within. Specifically, a ring is a PID if it is an integral domain (that means, among other things, it has no divisors of zero) and every ideal is principal. What does that mean?

A principal ideal is simply an ideal generated by a single element. For example, in a PID, if you pick any ideal, you can find some element in the ring such that every other element in the ideal can be written as the product of that single element and some other element of the ring. Because of this property, PIDs are incredibly tidy – every ideal neatly boils down to one element. You might think of it as a way of organizing the elements within the ring that echoes the simplicity of having a single leader or a 'principal' element. This simple structure leads to many other interesting properties in the domains of algebra and number theory.

Prime Ideal

Moving on to another crucial term in ring theory, we encounter the prime ideal. A prime ideal is a specific kind of subset within a ring that helps us identify how the multiplication structure of the ring behaves. Much like prime numbers are the building blocks of the integers, prime ideals play a similar role in rings.

A prime ideal must satisfy two main conditions: First, it cannot be the whole ring, since then we wouldn’t really be setting anything apart as special. Second, if a product of two ring elements lies in the prime ideal, then at least one of those elements must be in the ideal itself.

Think of a prime ideal as having a sort of 'catching' property. If any products fall into it – that is, if you multiply two things together and the result is in the prime ideal – then one of the original things had to be inside it to begin with. It's like saying if a recipe (the product) is considered vegan (prime ideal), then the ingredients (factors) must contain at least one vegan component.

Maximal Ideal

Continuing our journey through the landscape of ring theory, let's delve into the concept of a maximal ideal. This is, in a way, an ideal with a bit of an ego – it's as big as it can possibly get without being the whole ring. But despite its size, it cannot be bulged or expanded without turning into the entire ring itself.

Formally speaking, an ideal is maximal if it is properly contained within the ring (meaning it's not the entire ring), and there are no other ideals sitting comfortably between it and the full ring. If you try to include any more elements to a maximal ideal, you'll find it swelling up to become the whole ring. It’s like a balloon that's blown up as much as possible without popping – the maximal amount of air before it becomes the surrounding atmosphere.

The special thing about PIDs is that every non-zero prime ideal is also maximal. Showing why this is the case involves a bit of clever thinking about how elements and ideals interact in a PID.

Complex Numbers Norm

Complex numbers are a fascinating area of mathematics where we extend the familiar real number line into an entire plane. These numbers have both a real and an imaginary component, and just like real numbers, we can talk about their size or 'norm'.

The norm of a complex number is a non-negative real value that corresponds to the square of its distance from the origin in the complex plane. For a complex number \(\alpha = a + bi\), the norm, denoted as \(N(\alpha)\), is given by \(\alpha \bar{\alpha} = a^2 + b^2\), which is the sum of the squares of the real and imaginary parts. This resembles the Pythagorean theorem where the norm is the squared length of the hypotenuse of a right triangle whose sides are lengths |a| and |b|.

Why do we care about norms? They come in handy when we want to have a solid grasp of the 'size' or 'magnitude' of complex numbers. Knowing the norm allows us to compare complex numbers, talk about their convergence, and it plays a crucial role in various areas of higher mathematics, including number theory and analysis.

Ring Theory

Finally, let's take a step back and look at the broader picture: ring theory. Ring theory is the study of rings – mathematical structures consisting of a set equipped with two binary operations that generalize the arithmetic of integers. In rings, we can add, subtract, and multiply, but not necessarily divide.

Within ring theory, we analyze how different parts of the ring interact, like ideals. You can think of ring theory as examining a society. The elements are its people, the operations (like addition and multiplication) are the social interactions, and the ideals are the various social groups or communities within it. Some groups might have particular influences – like prime and maximal ideals – and understanding the structure and hierarchy of these groups helps us comprehend the society as a whole.

Ring theory is more than an academic exercise. It forms the backbone for vast areas of mathematics and has applications in computer science, physics, and beyond, providing a foundation for understanding complex systems in an organized and structured way.