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!