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:
- 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?
- And why are the ideals not present in the case of groups?
- 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.
Best Answer
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:
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.