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 $\mathbb Z[i\sqrt 3]$.
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!
Best Answer
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.