So I've been trying to study ring theory using online resources and it has been quite challenging. I really want to learn more about rings but it has been insanely frustrating so I'd like to ask if what I've learned is correct or if I am misunderstanding something.
Let R be a ring, $a,b\in R$
1: an ideal $(a)$ in $R$ is a subring that is closed under addition from any other element in itself and is closed under multiplication by any constant in $R$. $(a)$ can also be represented by the set $\left \{ ar \right. : r \in R\left. \right \}$. The ideal $(a)$ would be called principal because it is generated by a single element
2: an element of the ideal $(a,b)$ is of the form $am+bn$ where $b,m \in R$. This ideal is not principal (unless it simplifies to a principal ideal, like if $R$ were the ring of integers it would be principal)
3: The ideal norm $N(a)=\left | R/(a) \right |$. In the ring of imaginary quadratic numbers is there some formula to calculate the ideal norm? Ive seen somewhere that if $(a,b)=1$ (coprime), then $N(a+bi)=a^{2}+b^{2}$. Is there a generalization of this?
4: Ideal multiplication I'm shaky on. From intuition here is what I think: $(a)(b)=(ab)$. If my intuition is correct, then $(a,b)(c,d)=(ac,ad,bc,bd)$. From this logic, I'd think that $(a)+(b)=(a,b)$ but I dont think this is right
5: In a quotient ring $R/(a,b)$, $a=0$ and $b=0$
Thanks a lot for any help! Any resources (clear) would be very much appreciated aswell.