Intuition behind "ideal"

To briefly put forward my question, can anyone beautifully explain me in your own view, what was the main intuition behind inventing the ideal of a ring? I want a clarified explanations in these points:

1. Why is the name "ideal" coined?. In English 'ideal' means "One that is regarded as a standard or model of perfection or excellence." Why did people gave the name of ideal to such group?
2. And why are the ideals not present in the case of groups?
3. And give me a very fantastic intuition and motivation behind the ideals and what are the roles served by them in advanced mathematics.

Thanks a lot, to every one.

As was already said, the term "ideal" came from Kummer's ideal numbers (more precisely, "ideal complex numbers" as Kummer was concerned with factorizations of algebraic integers which lie in the complex field). I'll try to give a brief intuition not mentioned here explicitly already.

When factoring, say, 60, you find 2 "different" factorizations: $60=15\times 4=12\times 5$. This does not contradict unique factorization in integers since you have not factored "enough": after factoring all the numbers as much as possible you obtain the unique factorization $60=2\times 2\times 3\times 5$.

However, in the context of algebraic integers this does not always hold. The famous example is in $\mathbb{Z}[\sqrt{-5}]$. There you have $6=2\times 3=(1-\sqrt{-5})\times(1+\sqrt{-5})$ but the factors are irreducible, so unique factorization fails.

Kummer's idea was that in this case as well the problem is that the factors were not factored "enough". His approach was to assume that there are better, "ideal" factors for which the unique factorization hold.

It is obvious that there is something problematic here - you need to construct such ideal numbers, prove their existence, etc. However there is also another way created by Dedekind. Dedekind defines not the ideal numbers themselves, but the sets of elements they divide. For example, instead of talking about "2" you can talk about the set $\{0,2,-2,4,-4,6,-6,\dots\}$ of even numbers in the integer - the ideal created by 2.

Dedekind noted that this concept of "being divided by" can be characterized by two properties:

1. If some number (ideal or not) divides $a$ and $b$, it divided $a+b$.
2. If some number (ideal or not) divides $a$, it divdes $\lambda \times a$ for all $\lambda$.

So he defined ideals using these two properties. They turned out to be just enough to prove the grand theorem that is in the base of algebraic number theory - that in Dedekind domains (and so in algebraic integer ring) there is a unique factorization of elements of ideals to products of prime ideals (this applies to elements as well, since an element can be identified with the ideal it generates).

This is quite orthogonal to the usage of ideals usually encountered in an undergrad level algebra course - where ideals pop up naturally (and more generally) as kernels of homomorphisms. But here the name "ideal" is indeed confusing.

1. "Ideal" is meant in the sense of "not real." The story as I know it is that Kummer was interested in fixing unique factorization in number rings, and to do it he had to introduce certain "ideal numbers." Dedekind later recast these ideas in terms of ideals as we now understand them.

2. To me ideals are kernels of ring homomorphisms. The analogue of ideals for groups is normal subgroups.

3. Again, ideals are kernels of ring homomorphisms. In number theory, ideals appear naturally as the correct objects to use to study divisibility. After all, the statement that $a$ divides $b$ is nothing more than the statement that $b$ is contained in the ideal $(a)$ generated by $a$, or in the language of ring homomorphisms that $b = 0$ in the quotient ring $R/(a)$. So instead of studying divisibility of numbers we can study containment of ideals. For $\mathbb{Z}$ this gives us the same theory of divisibility, but for other number rings we get non-principal ideals which are precisely the correct manifestation of Kummer's "ideal numbers."

The notion of an ideal number was introduced by Ernst Kummer in an attempt to explain and fix the failure of unique factorization in certain subrings of the complex numbers. The canonical example that students usually see is that $2\cdot 3 = (1+\sqrt{-5})(1-\sqrt{-5})$ in $\mathbb{Z}[\sqrt{-5}]$.

It's fairly easy to surmise from that Wikipedia article, that working with ideal numbers was rather cumbersome. So where did the idea of considering ideals come from? I can only make an educated guess, but I think it was the same as with many other things that we now take for granted: It came from people looking at examples and eventually figuring out what it was that made them all "tick" - as it were.

So why are ideals useful? Because rings of algebraic integers in number fields are (generally) Dedekind domains where unique factorization - not in the ring itself, but rather of its ideals into its prime ideals holds.

For a long series of videos of lectures on abstract algebra ending up with some algebraic number theory about algebraic integers in imaginary quadratic number fields, there is this series of lectures from Harvard.