A Universe of Sorts

§ Prime numbers as maximal among principal ideals

I learnt of this characterization from benedict gross's lectures, lecture 31 . We usually define a number

p \in R

as prime iff the ideal generated by

p

(p)

is prime. Formally, for all

a, b \in R

, if

ab \in (p)

then

a \in (p)

b \in (p)

. This can be thought of as saying that among all principal ideals, the ideal

(p)

is maximal: no other principal ideal

(a)

contains it.

So we are saying that if $(p) \subseteq (a)$ then either $(p) = (a)$
Since $(p) \subseteq (a)$ we can write $p = ar$ . Since $(p)$ is prime, and $ar = p \in (p)$ , we have that either $a \in (p) \lor r \in (p)$ .
Case 1: If $a \in (p)$ then we get $(a) \subseteq (p)$ . This gives $(a) \subseteq (p) \subseteq (a)$ , or $(a) = (p)$ .
Case 2: Hence, we assume $a \not \in (p)$ , and $r \in (p)$ . Since $r \in (p)$ , we can write $r = r'p$ for some $r' \in R$ . This gives us $p = ar$ and $p = a(r'p)$ . Hence $ar' = 1$ . Thus $a$ is a unit, therefore $(a) = R$ .