Prime ideals

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Given a ring $R$, a prime ideal ${\mathfrak {P}}$ is a non-trivial ideal such that for any two numbers $m,n\in R$, if $mn\in {\mathfrak {P}}$, then either $m\in {\mathfrak {P}}$ or $n\in {\mathfrak {P}}$, maybe both.

For example, $\langle 7\rangle$ is a principal ideal and a prime ideal in $\mathbb {Z}$. For any $m,n\in \mathbb {Z}$, if it is the case that $mn\in \langle 7\rangle$, then either $m\in \langle 7\rangle$ or $n\in \langle 7\rangle$, maybe both.

For contrast, note that $\langle 21\rangle$ is a principal ideal in $\mathbb {Z}$ but not a prime ideal, for although $63\in \langle 21\rangle$, we see that 63 = 7 × 9 but $7\not \in \langle 21\rangle$ and $9\not \in \langle 21\rangle$ either. In $\mathbb {Z}$, only prime numbers generate prime ideals (this includes negative prime numbers like –47 and –109). Composite numbers don't generate prime ideals in $\mathbb {Z}$ but they do generate principal ideals.

A ring may lack unique factorization but it still has prime ideals, and all non-trivial ideals can be factorized uniquely into prime ideals. For example, $\langle 7\rangle$ is a principal ideal but not a prime ideal in $\mathbb {Z} [{\sqrt {-5}}]$. Verify that $21\in \langle 7\rangle$, that $(4-{\sqrt {-5}})(4+{\sqrt {-5}})=21$ and that $(4-{\sqrt {-5}})\not \in \langle 7\rangle$ and $(4+{\sqrt {-5}})\not \in \langle 7\rangle$. The ideal $\langle 3,1+{\sqrt {-5}}\rangle$, on the other hand, is a the of the [FINISH WRITING]