Ideals

A while ago, I wrote about unique factorization, and how it can sometimes fail. Today I want to introduce one concept that to some degree is intended to get around the failure of unique factorization in some environments: ideals.

There’s a fairly standard definition of ideals, which most books start from; here I’d rather explain how ideals arose.

If you just want to look at the multiplicative structure of a ring, for example if you want to study factorization, it makes sense to identify associates, to avoid having to distinguish, say, 2 from -2. The main step here is that we can look at sets like {-2, 2}, {-3, 3}, etc. on their own, or we can look at a fairly natural construction: the set of all even numbers, the set of all numbers divisible by 3, and so on. We’ll denote these sets with parentheses: (2), (3), (4)…

We can naturally multiply ideals. (2) times (3) is (6), since every even number times a number divisible by 3 will yield a number divisible by 6. We can factorize ideals: in the ring of rational integers, we can simply write something like (18) = (2)(2)(3), with no need to deal with associates.

We can also define ideals abstractly, by two crucial properties: first, we can add and subtract in an ideal, and second, we can multiply by any element of the entire ring. The sum and difference of two even numbers is even, and every even number times any integer is again even.

In the ring of rational integers (which I’ll call Z, like every serious mathematician), ideals have a very simple form: every ideal is simply the set of all numbers divisible by some number. The ideal (0) includes just 0, and the ideal (1) includes the whole ring; less trivial ideals are (2), (3), (4), etc.

However, in some rings, in particular in all rings without unique factorization, this is not the case: we can always find some ideal that isn’t the set of all numbers divisible by one element. For example, take the SQRT(-5) extension of Z, denoted Z[SQRT(-5)]: it has an ideal denoted (2, 1-SQRT(-5)), consisting of all elements of the form 2x + (1-SQRT(-5))y.

In Z, we can have things like (2, 4), (4, 6), or (2, 3). But in fact, we can distill them to one element: (2, 4) = (2), since every integer of the form 2x + 4y can be written as 2z, z = x + 2y; similarly, (4, 6) = (2), and (2, 3) = (1). In contrast, (2, 1-SQRT(-5)) is not like that. Unlike 4 and 6, 2 and 1-SQRT(-5) have no factors in common, except units; and unlike 2 and 3, it’s impossible to express 1 as a linear combination of the two elements.

We can still multiply ideals generated by more than one element, mind you. For example, in Z[SQRT(-5)], we have (2, 1-SQRT(-5))(3, 1-SQRT(-5)) = (6, 1-SQRT(-5)) = (1-SQRT(-5)) since 6 = (1-SQRT(-5))(1+SQRT(-5)). Even when normal factorization is not unique, factorization into ideals is in fact unique: (6) = (2, 1-SQRT(-5))(2, 1-SQRT(-5))(3, 1-SQRT(-5))(3, 1+SQRT(-5)), and all four ideals on the right are irreducible in the sense that they can’t be written as products of other ideals except themselves and (1).

By the way, note that bigger ideals correspond to smaller numbers. Intuitively, 2 and 3 are smaller than 6, and 2*3 = 6. But as ideals, (2) and (3) are bigger than (6), since (6) is a subset of both (2) and (3). Similarly, (2, 1-SQRT(-5)) and (3, 1-SQRT(-5)) are bigger than (1-SQRT(-5)).

Slight complication: the set of unique factorization domains, characterized by unique factorization without the need to resort to ideals, is strictly smaller than the set of principal ideal domains, characterized by the fact that all of their ideals are generated by one element. The ring Z[x], consisting of all polynomials with integer coefficients, has unique factorization, but has the ideal (2, x). However, if we only consider rings generated by attaching roots of various kinds to Z, then in fact all UFDs are PIDs.

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