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3.2 Principal Ideal Domains

Definition 3.2.1 A principal ideal domain (PID) is an integral domain in which every ideal is principal.

Lemma 3.2.2 Z is a PID.

NOTE: Showing that Z is a PID means showing that if I is an ideal of Z, then there is some integer n for which I consists of all the integer multiples of n.

Proof: Suppose that I  Z is an ideal. If I = {0} then I is the principal ideal generated by 0 and I is principal. If I ≠ {0} then I contains both positive and negative elements. Let m be the least positive element of I. We will show that I = (m).

Certainly (m)  I as I must contain all integer mulitples of m. On the other hand suppose a ∈ I. Then we can write

a = mq + r

where q ∈ Z and Principal Ideal Domain | Algebra - Mathematics Then r = a - qm. Since a ∈ I and -qm ∈ I, this means r ∈ I. It follows that r = 0, otherwise we have a contradiction to the choice of m. Thus a = qm and a ∈ (m). We conclude I = (m).

Note: In fact every subring of Z is an ideal - think about this.

Lemma 3.2.3 Let F be afield. Then the polynomial ring F[x] is a PID.

NOTE: Recall that F[x] has one important property in common with Z, namely a division algorithm. This is the key to showing that F[x] is a PID.

Proof: Let I  F[x] be an ideal. If I = {0} then I = (0) and I is principal. If I ≠ {0}, let f(x) be a polynomial of minimal degree m in I. Then (f(x))  I since every polynomial multiple of f(x) is in I.

We will show that I = (f (x)). To see this suppose g(x) ∈ I. Then

g(x) = f(x)q(x) + r(x)

where q(x),r(x) ∈ F[x] and r(x) = 0 or deg(r(x)) < m. Now

r(x) = g(x) — f(x)q(x)

and so r(x) ∈ I. It follows that r(x) = 0 otherwise r(x) is a polynomial in I of degree strictly less than m, contrary to the choice of f(x).

Thus g(x) = f(x)q(x), g(x) ∈ (f(x)) and I = (f(x)).

Question for the Seminar: If R is a ring (not a field) it is not always true that R[x] is a PID.

Find an example of a non-principal ideal in Z[x].

The document Principal Ideal Domain | Algebra - Mathematics is a part of the Mathematics Course Algebra.
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FAQs on Principal Ideal Domain - Algebra - Mathematics

1. What is a principal ideal domain in mathematics?
A principal ideal domain (PID) is a type of commutative ring in abstract algebra that has certain properties. Specifically, a PID is an integral domain where every ideal is generated by a single element. In other words, for any ideal I in a PID, there exists a single element a such that I is the set of all multiples of a.
2. What are the key properties of principal ideal domains?
The key properties of principal ideal domains are: - Every ideal is principal, meaning it is generated by a single element. - Principal ideals are closed under addition and multiplication. - Principal ideals have a unique factorization property, meaning every nonzero non-unit element of the PID can be expressed as a product of irreducible elements, up to order and unit multiples.
3. How does a principal ideal domain differ from other types of rings?
A principal ideal domain differs from other types of rings, such as Euclidean domains or polynomial rings, in terms of the properties it possesses. While Euclidean domains have a division algorithm and polynomial rings have polynomial division, principal ideal domains do not necessarily have a division algorithm. However, they do have the property of every ideal being principal, which is not necessarily true for other types of rings.
4. Can every integral domain be a principal ideal domain?
No, not every integral domain can be a principal ideal domain. While every principal ideal domain is an integral domain, there are integral domains that do not satisfy the property of having every ideal be principal. Examples of integral domains that are not principal ideal domains include polynomial rings in more than one variable and certain rings of algebraic integers.
5. How are principal ideal domains useful in mathematics?
Principal ideal domains are useful in various areas of mathematics, particularly in algebraic number theory and commutative algebra. They provide a rich structure for studying factorization properties of elements in a ring, allowing for unique factorization of elements into irreducible elements. This is important in proving results related to prime numbers and prime factorization. Additionally, principal ideal domains are used in constructing quotient rings and studying algebraic structures like modules and vector spaces.
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