## PIDs (Pt. II)

**Point of Post: **This is a continuation of this post.

We now would like to discuss the notion of greatest common divisors in PIDs. Recall that the notation , *divides* , for in some ring is that there exists some such that . Note that this is equivalent to saying that . Now, note that if is an integral domain and and , say and we see then that and so by cancellation so that . We call two elements related in this way, and , *associated* and we call them *associates.* It’s evident that being associate is an equivalence relation on . We then call an element a *greatest common divisor *of if , and whenever and then . An emphasis was placed on the article ‘a’ because greatest common divisors are not unique. For example, in a greatest common divisor of and is , but so is . In general, greatest common divisors are unique up to associates:

**Theorem: ***Let be an integral domain and let . Then, if is a greatest common divisor of then so is for any . Moreover, every greatest common divisor is of this form.*

**Proof: **Suppose first that is an associate of , then evidently and if then so that and so is a greatest common divisor of . Conversely, suppose that is a greatest common divisor of . Since and is a greatest common divisor we have that . But, since is also a greatest common divisor of applying the same argument shows that . From previous observation we have that and are associates.

So, how exactly do greatest common divisors factor into PIDs? Well, given two elements we know that the ideal generated by the two is principal, and so equal to for some . Well, unsurprisingly, as is easily seen in , one has that is a greatest common divisor of . Indeed:

**Theorem: ***Let be an integral domain and suppose that are such that . Then, is a greatest common divisor of .*

**Proof: **Since we have by definition that . Suppose now that then we know that and so . But, by definition this implies that .

So, think about how this jives with our usual notion of greatest common divisors in . We usually call the greatest common divisor of the largest (positive) common divisor of . In other words . Now, from 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 particular, . From this it’s not hard to conclude that, agreeing with our idea of greatest common divisors in , one has the following:

**Theorem: ***Let be a Euclidean domain with satisfying the -inequality. Then if one has that is a greatest common divisor of if and only if and whenever one has that .*

**Proof: **Suppose first that is a greatest common divisor of and let . As mentioned in the previous paragraph we have that and so and so . We shall prove the other direction in our next post.

Regardless, another common fact (known colloquially as Bezout’s identity) which holds true in holds more generally in PIDs:

**Theorem: ***Let be a PID. Then, if one has that is a greatest common divisor of if and only if for some .*

**Proof: **Suppose first that have as a greatest common divisor. We know then that and so, in fact, for any there exists such that . Conversely, suppose that for some then evidently and so is a greatest common divisor of .

If have as a greatest common divisor we say that are *coprime*. In fact, we see that the above theorem may be stated as follows:

**Theorem: ***Let be a PID and . Then, are coprime if and only if are comaximal. *

And so the CRT tells us that if are coprime then .

As a last structural result we mention the following:

**Theorem: ***Let be a PID, then is a PID if and only if .*

**Proof: **The necessity for being prime is clear since we need to be an integral domain. Suppose now that is prime, then is an integral domain. Now, if is an ideal in then we know by the lattice isomorphism theorem that for some . But, by assumption that is a PID we know that for some and it’s not hard to see then that .

**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.

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