Naïve groups, fields and ideals Please excuse the simplicity of this question, but I am very new to groups and fields. I only seek an simplistic / intuitive expalnation, and confirmation / refutation re whether I am on the right track. I confess I am a little averse to reading about them from the ground up, as my motivation is quickly deteriorated when goin gthourough rigourous set theoretic foundations.I could put this down to different learning styles at best, laziness at worst! But I feel I learn best from concrete examples and  pictoral representation - I am not at the abstract level of understanding yet really.
Rading Derbeshire's "Unknown quantity" he gives an example of and ideal in $\mathbb{N}$ as $\dots -60,-45,-30,-15,0,15,30,45,60\dots$
Does this extend to a quadratic ideal eg $\dots -32,-16,-4,-8,-2,0,2,4,8,16,32 \dots?$
Is it really this simnple, or am I missing something crcial, or have I got completely the wrong end of the stick? I am sure that there is much more to it than this, but is this a sound starting point to beginning to understand the modular nature of these things?
 A: While that's actually a pretty clever idea, it's not what ideal means. First of all, Ideals need to be subrings, so if $16$ and $-4$ are in the ideal, then so must $16+(-4)=12$. Secondly, Ideals are meant to be "closed under multiplication." The idea is to generalize the notion of "has a factor of n." So, for any number $n$, we know that $n$ times something in (⋯−60,−45,−30,−15,0,15,30,45,60…) will have a factor of 15. The canonical example is even numbers. Anything times an even is an even. So the ideal $(...,-6,-4,-2,0,2,4,6,...)$ is the example you want to keep coming back to. The "quadratic ideal" your suggesting doesn't have this nice property, sadly.
This is a great question, by the way.
A: You have an interesting thought. However, if you know about Normal Subgroups, you know that the kernel of a homomorphism is worth naming and study. An Ideal is useful because it is the Kernel of a homomorphism from one Ring to another.
This means an Ideal has to relate to both the additive and multiplicative structures. You have tried to capture the multiplicative idea, but, for example $2+4=6$ is not in your set, so it doesn't work for addition.
The work you put in at this stage in understanding will definitely pay off, so do keep asking questions until you have got it clear.
