Hi I am wondering if someone can explain something in my notes.
It says, consider $$\mathbb{Z}(\sqrt{5})=\{a+b\sqrt{5}:a,b \in \mathbb{Z}\}$$
then $$I=\{5a+b\sqrt{5}: a,b \in \mathbb{Z} \}$$ is an ideal. Without verifying axioms, this can be seen by noting it is simply the principal ideal $(\sqrt{5})$
My question is; because I am rather new to rings so I am a bit confused.
I know a principal ideal is something of the form $(n)=(rn)$ where $r \in R$ i.e. just basically all the multiples under the multiplication ring operation.
So for example $$(5)=(-,,..-5,0,5,10,15,…)=\{5n: n\in \mathbb{Z}\}$$
so $(\sqrt{5})=(\sqrt{5}n)$
so where does the extra $5a$ come from/ why is that allowed? it doesn't change the ideal?
An ideal can be generated by more than one element. In your case, $$ I = (5, \sqrt{5}) = \{ 5a + \sqrt{5}b \mid a,b \in \Bbb{Z} \}. $$ Even though it's presented as generated by two elements, it turns out in this case that the ideal is generated by a single element (that's what principal means). What does it mean to be generated by an element $x$? It means that any element of $I$ can be written as a multiple $rx$, where $r$ can be any element of the ring. Since $$ 5a + \sqrt{5}b = ( b + \sqrt{5}a ) \sqrt{5}, $$ every element of $I$ is a multiple of $\sqrt{5}$. Thus, $$ I = (\sqrt{5}). $$