## Euclidean Domains (Pt. I)

**Point of Post: **In this post we set the stage, in the typical manner, for the decreasing sequence of ‘nice integral domains’ starting from the nicest (non-field), Euclidean domains.

*Motivation*

Now that we have ample information about general rings, it would seem to behoove us to start looking of classes of better behaved rings (similar to looking at Hausdorff spaces for topological spaces). In a somewhat surprising, but standard, approach we start in the opposite direction from where one might expect. In other words, we shall start with the nicest class of rings we shall consider and work our way down to the least nice, but most general. So, what is this class of nice rings for which we start? Well, anyone who has done any kind of number theory knows that the inability of division in is not as hard an obstacle for one to overcome as may be initially thought (of course, this inability to divide is what makes number theory a rich subject). Indeed, this is the case because while doesn’t have division, it has a *division algorithm*. Given any two integers we know that we can write for some integer and some number . The operative part, at least for us, is that we know we can make the remainder be ‘small’ (compared to ) since this allows us to perform minimality arguments (i.e. every subgroup of is generated by the element of least norm). This is the idea we’d like to abstractify. Namely, we’d like to consider classes of integral domains where we have a notion of ‘size’ which allows us to create a ‘division algorithm’ where the remainder has small ‘size’. In particular, this will entail us having an integral domain for which there is a ‘size function’ such that for every there exists such that where .

*Euclidean Domains*

A *norm* on an integral domain is a function with the property that . Norms are, as goes without saying, not very interesting. Some more interesting qualities a norm can have is that it may be *multiplicative *or positive for . It’s clear actually, that only integral domains (or more precisely, rings without zero divisors) can have positive multiplicative norms, for if then and so , but if then and so contradictory to what was previously proven. Conversely, if is an integral domain then defining gives a positive multiplicative norm on . Thus, one could phrase that integral domains are precisely those commutative unital rings which admit positive multiplicative rings.

Norms aren’t what ultimately interests us right now though, not even positive multiplicative norms. No, we are interested in a special type of norm called a Euclidean valuation. In particular, a norm on an integral domain is called a *Euclidean valuation *if given any with there exits with such that either or . Such a are called a *quotient *and *remainder* respectively. It is standard practice to call the value the *degree of .*

So, why is this function useful? Well, the first thing it allows us to do is perform minimality arguments related to division. To illustrate consider the following proof that every ideal of the Euclidean domain is principal (generated by a single element of ):

**Theorem: ***Let be a Euclidean domain with Euclidean valuation, then if is an ideal of then where is any element of of minimal degree. *A *Euclidean Domain* is an integral domain for which there exists a Euclidean valuation on .

**Proof: **First of all, it’s clear by the well-ordering principle that there exists elements of of minimal degree. Suppose that is such an element. Clearly and so it suffices to prove the reverse inclusion. To do this let be arbitrary. We know then, by definition, that there exists with with or . That said, since and has minimal degree in we have that cannot be the case, and so . Thus, , or . Since was arbitrary we have that and the conclusion follows.

Ok, fine, so we can see that an integral domain being Euclidean has its perks (this ‘principal ideal property’ is actually just one small fact), but for the notion of Euclidean domains to be useful, they must be common enough that we’d care. In other words, what are some examples of Euclidean domains? Well, probably the most obvious examples besides fields (where one can just define for all ) is with . The fact that is, in fact, an integral domain was proven to you before, but it was just called the ‘division algorithm’. Other important examples include the quadratic integer ring with . This is, in fact, not hard to show that is a Euclidean domain where is some field . These three examples will constitute a lot of our time in the future, and we shall prove that they are Euclidean domains when their time comes. I should point out, that although the focus of the definition of Euclidean domains is on the function, probably the hardest part, which is implicitly implied in the definition is showing first off that every element can be written in the form .

So, what are some non-examples of Euclidean domains? Well, while a lot of the theory that we shall see for Euclidean domains will be developed in later posts I discussed the fact that all ideals in Euclidean domains are principal because it gives a good way of proving that a given integral domain is not Euclidean. Indeed, it’s easy to see that the ring is not Euclidean. Why? Well, I claim that the ideal for any prime is not principal in . Why though? Well, suppose for a second that it was, that for some polynomial . Since this implies that for some , and so, in particular is constant. But, this then implies that is also constant, and since divides this implies that or . Well, clearly otherwise which would contradict . Thus, we may conclude that but, evidently which is again a contradiction. Thus, is not principal in and so cannot be Euclidean (note, this means that there does NOT EXIST a which makes into a Euclidean domain–not that it isn’t a Euclidean domain for a specific function)

**References:**

[1] Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. *Advanced Modern Algebra*. Providence, RI: American Mathematical Society, 2010. Print.

[3] Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. *Basic Abstract Algebra*. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.

[…] this post we discuss a more general class of rings than Euclidean domains. The type of ring we are talking about is a natural one to consider. In particular, we have seen […]

Pingback by PIDs (Pt. I) « Abstract Nonsense | October 21, 2011 |

[…] the previous theorem we know that if and then and so, in particular, we have that . But, by previous theorem we know is generated by an element of least degree in . In particular, for every and so, in […]

Pingback by PIDs (Pt. II) « Abstract Nonsense | October 22, 2011 |

[…] this we point we have discussed the very nice integral domains which admit degree functions (i.e. Euclidean domains) and the, less nice still fantastic, integral domains whose ideals are all principal (i.e. PIDs). […]

Pingback by UFDs (Pt. I) « Abstract Nonsense | October 22, 2011 |

[…] some in the study of . Well, it turns out that the set of all annihilating is an ideal. But, we know that is a PID and so this ideal is generated by some element of …I’ll give you three […]

Pingback by Modules (Pt. I) « Abstract Nonsense | October 27, 2011 |