202505211405
Tags : Ring Theory
Noetherian Ring and PIDs
Definition
A commutative ring is called a Noetherian Ring if every ideal is finitely generated.
The stronger version of the condition that forces every ideal to be generated by exactly 1 element is
Definition
A commutative ring is called a Principle Ideal Domain if every ideal is a Principle Ideals.
Example
is a PID. This explains why greatest common divisors behave the way they do, If one wants the set of all elements that are divisible by both , then one can find an element that generates it, which is the ideal generated by the greatest common divisor.
Example
If is a field, then , i.e the ring of polynomials, is a PID, which can be shown easily by just the euclidian algorithm for finding the HCF of 2 numbers.