Examples of prime-like entities on rings other than natural numbers

Question

For the natural numbers, primes can be viewed as the building blocks.

Are there other significant examples in rings other than $\mathbb{N}$ (that do not have a one-to-one mapping with $\mathbb{N}$) that has something similar to primes (that are not necessarily "numbers")?

By significant I mean the prime-like stuff plays an important (at least interesting) role in the structure of that ring.

I guess the ring itself would have to be non-trivial or "complex enough" to have such an inner-structure.

Answer 1

This is a general thing that is studied for any commutative ring and has interesting structure associated to it.

First, let us assume the ring is an integral domain, since this makes everything much nicer and fits in with the main examples.

Then we need to observe that there are two "natural" ways to define what it means to be like a prime number. For natural numbers, we know that a number $p$ being a prime (meaning that the number cannot be factored further) is the same as the condition that whenever $p$ divides a product, then $p$ divides one of the factors.

These conditions turn out to not be equivalent for integral domains in general, so we call elements satisfying the first condition (and which are neither $0$ nor invertible) irreducible, whereas we call elements satisfying the second condition (still being neither $0$ nor invertible) prime elements.

Now, to build a "good" theory similar to prime factorization in the integers, we need to have all elements be factorizable in a unique way, which turns out to imply that the two notions above coincide.

Once these notions do coincide, and if we assume that all elements can be factored into irreducible elements (which is the case for most rings one encounters "naturally"), we indeed do get that each element can also be factored uniquely into irreducible elements, just like we are used to.

Having unique factorization is important in many examples. It holds in the ring of Gaussian integers $\mathbb{Z}[i]$, leading to a formula for the number of ways to write an integer as a sum of two squares. It also holds in the ring of integers with a third root of unity added, leading to a proof of Fermat's last theorem for the $n=3$ case.

(Failure of unique factorization was what lead to an early attempted proof of Fermat's last theorem being false, and to the theory of Dedekind domains and factorization of ideals in those).

Answer 2

A common example is polynomial rings, i.e. the set of all polynomials, whose coefficients come from some field (e.g rationals or reals). The primes are the polynomials that can't be factored, e.g. $x-c$ for any constant $c$.