PRINCIPAL IDEAL DOMAIN

A 'principal ideal domain', or 'PID', is a type of mathematical object which is part of the broader type of objects known as ''rings''. A ring admits operations of addition and multiplication, the archetypal example being the familiar ring of integers. In considering rings, one of the first questions which arises is whether elements can be
factored uniquely in an appropriate sense. The question is hard to even state if multiplication is not ''commutative'', that is, if the result of a multiplication depends on the order of the factors. Furthermore, if there are non-zero elements that multiply to zero, called ''zero divisors'', then there is no prospect of achieving any such uniqueness. A ring in which multiplication is commutative, and in which there are no zero divisors, is called an ''integral domain'', and the attention of factoring is naturally restricted to them.
There are strong relationships between questions of factorization of its elements and the structure of certain subsets of a ring called ''ideals'': these are the non-empty subsets of a ring which are closed under addition by any element of the ideal, and under multiplication by any element of the entire ring. For instance, the even numbers are an ideal of the ring of integers, since the sum of an even number with another even number is even, and the product of an even number with ''any'' integer is even. This ideal consists of the multiples of the number 2, but some rings contain ideals which are not generated by a single element in this way. An ideal is called ''principal'' if it is generated by a single element, and an integral domain is called a ''principal ideal domain'' if all its ideals are principal.

Contents

Examples

Modules

Properties

References

Examples

Examples include:

★ ''K'': any field,

★ 'Z': the ring of integers,

★ ''K[x]'': rings of polynomials in one variable with coefficients in a field.

★ 'Z'[''i'']: the ring of Gaussian integers

★ 'Z'[ω] (where ω is a cube root of 1): the Eisenstein integers
Examples of integral domains that are not PIDs:

★ 'Z'[''x'']: the ring of all polynomials with integer coefficients.
It is not principal because the ideal generated by 2 and ''X'' is an example of an ideal that cannot be generated by a single polynomial.

★ ''K[x,y]'': The ideal ''(x,y)'' is not principal.

Modules

The key result here is structure theorem for finitely generated modules over a principal ideal domain. This yields that if ''R'' is a principal ideal domain, and ''M'' is a finitely
generated ''R''-module, then ''M'' has a minimal generating set, somewhat akin to a basis
for a finite-dimensional vector space over a field ( whose existence is, in fact, a special case of the result for general PIDs).

Properties

In a principal ideal domain, any two elements ''a'',''b'' have a greatest common divisor, which may be obtained as a generator of the ideal ''(a,b)''.
All Euclidean domains are principal ideal domains, but the converse is not true.
An example of a principal ideal domain that is not a Euclidean domain is the ring
ight]
^[1].
Every principal ideal domain is a unique factorization domain (UFD). The converse does not hold since for any field ''K'', ''K''[''X'',''Y''] is a UFD but is not a PID (to prove this look at the ideal generated by $leftlangle\; X,Y$
ight
angle. It is not the whole ring since it contains no polynomials of degree 0, but it cannot be generated by any one single element).
#Every principal ideal domain is Noetherian.
#In all rings, maximal ideals are prime. In principal ideal domains a near converse holds: every nonzero prime ideal is maximal.
#All principal ideal domains are integrally closed.
The previous three statements give the definition of a Dedekind domain, and hence every principal ideal domain is a Dedekind domain.
So that PID $subseteq$ Dedekind $cap$ UFD . However there is another theorem which states that any unique factorisation domain that is a Dedekind domain is also a principal ideal domain. Thus we get the reverse inclusion Dedekind $cap$UFD $subseteq$PID, and then this shows equality and hence, Dedekind $cap$ UFD = PID.

References

1. Wilson, Jack C. "A Principal Ring that is Not a Euclidean Ring." Math. Mag '46' (Jan 1973) 34-38 [1]