Could anyone provide me with very simple examples of prime ideals (that is,principle ideals in the ring of integers which are generated by a prime), explaining me the way they are generated? The public audience is made up of high schoold people, abour 17 years of age......
Thanks in advance.
Prime ideals in the integers are exactly the ideals generated by a single element. For example, you might take $(2) = 2\mathbb{Z} = \{ 2z \mid z \in \mathbb{Z}\}$ or $(3) = 3\mathbb{Z} = \{ 3z \mid z \in \mathbb{Z}\}$. Replace 2 or 3 with any prime $p$ and arrive at a similar notion.
To be more explicative, take $(7)$. This is the set of all integer multiples of 7, i.e. $\{\cdots,-14,-7,0,7,14,21,\cdots\}$. The reason this is an ideal is because if you take any integer and multiply it by an element of this list, you will get another element of the list, sometimes said it absorbs the ring of integers.
As per a comment, an ideal $P$ is called prime when given any two ring elements, say $r$ and $s$, if $rs \in P$, then either $r \in P$ or $s \in P$. For instance, if $a \cdot b = 7 \cdot c$ for some $a,b,c \in \mathbb{Z}$, either $7 \mid a$ or $7\mid b$, right? How else would 7 be a factor of the product?
Prime ideals are nice because you can use them to split up quotient rings by an ideal via the Chinese Remainder Theorem. Letting your ring be $R$, if $I = \prod_{j=1}^k I_j^{e_j}$ and $I_j$ are ideals and pairwise coprime (for any $m$ and $n$, $I_m + I_n = R$), with $e_j$ as positive integers, then $R/I \simeq \prod_{j=1}^k R/(I_j)^{e_j}$. That is to say, you can split up the quotient ring into quotient by these smaller ideals. For instance, taking $\mathbb{Z}/30\mathbb{Z}$, we know $30\mathbb{Z} = 2\mathbb{Z}\times 3\mathbb{Z} \times 5\mathbb{Z}$, and each of these are pairwise coprime, we have that $\mathbb{Z}/30\mathbb{Z} \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z}.$