Euclidean rings and generators of ideals
This is to address a point that has just been glazed over in “Topics in Algebra” by Herstein.
In a Euclidean ring, for any two elements such that . Also, there exists a function such that .
We also know that the element with the lowest d-value generates the whole ring . The proof of this is elementary.
But what if there are more than one element with the same lowest d-value? Do both these elements generate ?
Yes. Proof: Let such that and are the elements of with the lowest d-value. Then for a third element , . Hence, both and divide . They also divide each other. Hence, and have to be associates. In other words, , where is a unit in .
Let us approach this from the opposite direction now. If , where is a unit. Axiomatically, . Hence, . Similarly, . This shows that . Therefore, whenever two elements and are associates, their d-values are the same.
Note that if we did not have the axiom that , then there would be no reason to believe that if and are associates, then . Hence, ideals could then potentially be generated by elements whose d-values would not be the lowest in the ideal, with the restriction that all those elements would be associates of the lowest d-value element.
A summary of the important points is:
1. Associates have the same d-value.
2. An element generates an ideal iff it has the lowest d-value in the ideal.
3. All associates of the lowest d-value element in an ideal generate the same ideal.
4. If we did not have the axiom , then point 1 would not be true, point 2 would not be true (a generator of an ideal wouldn’t have to have the lowest d-value), but point 3 would still be true.
There’s a couple of things I’d like to add here.
Why is it that a prime element should be such that its factorization does not contain a unit element? Generally, when we think about prime numbers in positive integers, we imagine a number which is absolutely not factorizable except in the form ( being the prime number). A sense of unbreakability is felt. Here, the same prime element can be broken in at least ways, where is the number of unit elements in the Euclidean ring . The sense of absolute unbreakability is lost. I suppose the reason for this is that the concept of ‘unit’ is just an extension of in natural numbers. As factorization of primes of the form are not counted when dealing with natural numbers, factorizations of the form shouldn’t count in , where is an associate of .
Also note that the addition of deletion of axioms would have greatly changed the structure of Euclidean rings. For example, deleting the axiom would allow infinite prime factorizations of elements, and adding the axiom would further alter the structure of . One should not forget that the properties of the elements of are a result of these defining axioms, and the addition and deletion of such would cause substantial alterations. It is just the fact that Euclidean rings mimic many properties of natural numbers that we find them important to study.