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Atiyah-MacDonald: Ideals June 18, 2010

Posted by Sarah in Uncategorized.
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An ideal, a, of a ring, A is a subset which is an additive subgroup and such that A a \subset a. The cosets of this ideal, A / a, are known as a quotient ring.

A basic, though special, example here is the ring of integers. Every ideal is the set of multiples of a single integer, n. The quotient rings are the integers mod n. The ideal generated by n is denoted (n), and it is called a principal ideal. Because all the ideals in the integers are multiples of a single element, the integers are known as a principal ideal domain.

Notice that (mn) \subset (n). Multiples of six are also multiples of three. An ideal generated by a prime number is not contained in any other ideal (except the whole ring of integers.) This brings us to the notion of prime and maximal ideals.

A prime ideal p in a ring A is an ideal that is not equal to (1) and if xy \in p then x \in p or y \in p. Equivalently, a prime ideal is one where A / p is an integral domain.

A maximal ideal m in a ring A is an ideal that is not equal to (1) and there is no ideal a such that m \subset a \subset (1). Equivalently, a maximal ideal is one where A / m is a field. (This last is true because A is a field iff the only ideals in A are 0 and (1).)

Maximal ideals are always prime, because a field is always an integral domain.

Every ring (not equal to the zero ring) has at least one maximal ideal. This is a result of Zorn’s lemma.

As a result, every non-unit of A is contained in a maximal ideal.

In a principal ideal domain, every non-zero prime ideal is maximal.
For if (x) \neq 0 is a prime ideal and (x) \subset (y), then we have some x = yz so that yz \in (x), hence z \in (x), say z = tx. Then x = yz = ytx, so yt = 1 and so (y) = (1).

The set of nilpotent elements in a ring is called the nilradical and it is an ideal. The product of anything with a nilpotent element is also nilpotent; the sum of nilpotent elements is nilpotent by the binomial theorem.

The nilradical is the intersection of all the prime ideals; if f is nilpotent, then f^n = 0 for some n, so f itself is in all the prime ideals. If f is not nilpotent, you can construct a prime ideal — the largest in the set of ideals a such that $f^n$ is not in a. (The product of two elements not in this ideal is not itself in this ideal.)

The Jacobson Radical is the intersection of all the maximal ideals in A. It (obviously) contains the nilradical.

You can get new ideals from old by summing them (that gives you a new ideal), intersecting them (also an ideal) or finding the product (the set of all finite sums \sum x_i y_i of products of elements in X and Y). In the integers, a \cup b is the ideal generated by their lcm, and $ab = (mn)$, the ideal generated by the product of their generators. If the generators are relatively prime, then the intersection and the product are the same ideal.

This is true for general rings, as can be proven by induction. The generalization of relatively primality is that two ideals are coprime if a + b = (1).

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