Intuition behind Irreducible Elements and Prime Elements

Question

The following is from C.Musili's Introduction to Rings and Modules.

Let $R$ be a commutative integral domain with unity.

Definition (Irreducible elements): A non-zero non-unit $a\in R$ is said to be irreducible if $a = bc$, then either $b$ or $c$ is a unit, i.e. $a$ cannot be written as a product of two non-units. Equivalently, the only divisors of $a$ are its associates or units.

Definition (Prime Elements): A non-zero non-unit $a\in R$ is said to be prime if $a|bc$ ($b,c\in R$), then either $a|b$ or $a|c$.

What is the intuition behind the two definitions above? I understand what they mean, and how they are different - but how were these definitions motivated? What should I think about when I think about irreducibility or primality (besides the definitions)?

In some sense, I thought irreducibility (by the English definition) meant that irreducible elements cannot be further broken down into "smaller" elements? The definition seems to say that irreducible elements are those non-zero non-units, which cannot be broken down as the product of two non-units. However, they can still be broken down into their associates or units - so what is the point? Are they really "irreducible"?

When learning about $\mathbb N$ as a child, I had similar notions of prime elements in mind - i.e. they are the building blocks of all numbers in that they cannot be broken down as the product of numbers except $1$ and the number itself. Perhaps my childhood intuition has misguided me at this stage since this appears to be what we mean by irreducibility? How do I think of primality, then?

I also know that primes are irreducible, but the converse is not true (follows from the definitions). I suspect this could be a reason why I am confusing the two notions (in terms of intuition; of course I understand the definitions and difference otherwise). An example of an irreducible element that is not prime is $1 + i\sqrt 3$ in $\mathbb Z[i\sqrt 3]$.

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Could you please give me intuition for irreducibility and primality (in the context of rings, or more generally if you wish) like I'm five? Thank you!

Answer 1

When we're dealing with factorisations, we need to be clear that we're talking about factorisation up to units.

Every integer has a unique factorisation. But we can write $6 = -2 \cdot -3$ just as well as $6 = 2 \cdot 3$. In order to consider these the same factorisations, we have to accept the fact that factorisation is really only defined up to reordering factors and multiplying factors by units.

Thus, an irreducible element is one which can only be factored by a unit and an associate.

It would be silly to say that $2$ is not irreducible because we can write $2 = -1 \cdot -2$, for example. Thus, we have to allow an irreducible to be factored by a non-trivial unit (such as $-1$ in $\mathbb{Z}$).

As for primality, there's not a nice intuition that I know of. It's just an incredibly useful concept.

Answer 2

To understand both, I think it is helpful to have a sense of where they came from, in terms of their history.

Both ideas came from number theory. The definition of prime numbers you learned as a child was developed first. Precisely, it is that a number $a$ is prime if, whenever you can write it as $a = bc$, then either $b = 1$ or $c = 1$. As number theory developed further, it was realized that another definition was valuable, namely the definition that $a$ is prime if, whenever $a | bc$, then either $a | b$ or $a | c$.

The reason why this is valuable is that much of elementary number theory is conceptualized in terms of this divisibility relationship, rather than in terms of factorization. The first definition says intuitively that prime numbers are the "primitive"/"smallest" in terms of factorization, whereas the second definition says intuitively that prime numbers are the "primitive"/"smallest" numbers in terms of divisibility. This is a subtle intuitive shift, but it is one that probably led to the generalization in terms of lattice theory, which nicely captures many of the central ideas of elementary number theory.

However, when people starting trying to generalize the concept of prime numbers to ring theory, they realized they had a problem. While the two definitions are equivalent in the context of number theory, they are not equivalent in ring theory. In addition, the first definition of primes has some slight issues.

The fix to the first definition is easy. Basically, where primes in number theory require that if $a = bc$, then either $b = 1$ or $c = 1$, now the only change is that $a = bc$ implies that either $b$ is a unit or $c$ is a unit. In other words, the concept of unique factorization in $\mathbb{N}$ needs to be generalized to the notion of unique factorization up to units. As Mark pointed out, this even happens when we generalize to $\mathbb{Z}$. Despite what it might seem, this is really only a very minor technical fix. However, it is not that easy to explain why. I would just recommend playing around with a few examples to understand it better.

The non-equivalence leads to a slightly larger headache. Which definition should we use? Well, the real solution to use both, but now have them be different concepts. The first definition of prime numbers is now called irreducibility. I imagine this is because the name fits well, but also because I believe the concept had already been introduced in the study of polynomials. Regarding the second definition, this became the notion of prime elements, and I believe the motivation here was that many of the ideas of number theory regarding divisibility very naturally apply to ring theory. That is, many of the spectacular aspects of prime numbers are captured by the concept of prime ideals, which are essentially ideals generated by prime elements.

Answer 3

For unique factorisation domains like the integers and $\mathbb Q[x]$, "prime" and "irreducible" coincide, so there is no confusion. The confusion only arises with non-UFD integral domains like $\mathbb Z[\sqrt{-5}]$.

All factors of an irreducible element are trivial, being units or associates. In that sense it cannot be "reduced" to a simpler expression. Primality is a stronger condition; it says that multiplication by that element can never "erase its legacy" (allow a factorisation that splits the element across several factors, which are then not divisible by the original element).