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Assume that $R$ is an integral domain. I want to prove that every finite subset $B$ has a greatest common divisor if and only if $B$ has a least common multiple. (As the (helpful) comment below shows, this is not immediately obvious.) I tried to use mathematical induction on the number of elements in $B$. If $B$ has 2 elements, say $b_1$, and $b_2$ then the identity $${\rm gcd}(b_1, b_2){\rm lcm}(b_1, b_2)={b_1}{b_2}$$ helps. The above identity can be proved using elementary divisibility arguments, and my checking it through seems to show that holds in an arbitrary integral domain. But even if $B$ has $3$ elements, the identity linking the product to the greatest common divisor, and least common multiple becomes complicated. Not sure how to set up the induction step. Thanks.

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but I am trying to answer the question in a general integral domain

Not possible, since there are examples of domains that don't admit GCDs. The special ones that do are called gcd domains and that is indeed equivalent to having lcms.

If two elements in an integral domain are coprime, can one write 1 as an R -linear combination of these elements?

No, for example, try to write $1$ as a combination of $x$ and $y$ in $\mathbb Z[x,y]$.

I tried to prove this by induction on the number of elements in B

I think all you need induction for, maybe, is to reassure yourself that the gcd condition you have works for all such $B$ iff it works for all pairs of elements. The same goes for lcms, then you can work with the much simpler pairwise conditions.

If the gcd of any subset B in an integral domain R can be written as a finite linear combination of the elements in B (with R coefficients), then R must be a PID. No idea how to prove this!

Neither do I, because this is also not true. A Bézout domain is a GCD domain in which you can write GCDs as a linear combination, but Bezout domains are not PIDs in general.

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  • $\begingroup$ If the gcd of any subset B in an integral domain R can be written as a finite linear combination of the elements in B (with R coefficients), then R must be a PID. No idea how to prove this! $\endgroup$ – student Jan 26 '18 at 4:25
  • $\begingroup$ @student I've added to my solution to address your comment. I don't know where you're getting these "facts" but my advice is to vet them more carefully in the future. Otherwise you'll lose a lot of time attempting to prove false statements. $\endgroup$ – rschwieb Jan 26 '18 at 17:06
  • $\begingroup$ This is an exercise in the textbook "A Course in Galois Theory" by D.J. H. Garling. Exercise 3.21, page 31. I seem to have figured it out. As far as I can make out, we consider any non-zero ideal J (say), and prove that the greatest common divisor d (say) generates J. One has to prove the set inclusions both ways and it seems straight-forward. $\endgroup$ – student Jan 27 '18 at 19:08
  • $\begingroup$ rschweib-your hint proved very helpful. I solved the question that I raised. I will post my solution when I have time. The question is also from the same textbook: A Course in Galois Theory by D.J.H. Garling. It is Exercise 3.19, page 30. $\endgroup$ – student Jan 27 '18 at 19:11
  • $\begingroup$ Your comment about Bezout domains makes sense. I think the hypothesis in the exercise in Garling's book is that R is a Bezout domain. He does not use this terminology, but to prove a very basic statement like the one in that exercise, I guess it is safe to omit the terminology. $\endgroup$ – student Jan 28 '18 at 2:40
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We argue by mathematical induction. If $B$ is a singleton $\{b\}$, then $b$ is both its own greatest common divisor, and its own least common multiple. So the existence of $b$ ensures that the existence of the gcd is equivalent to the existence of the lcm. If $n=2$, and $B$ is an arbitrary subset $B=\{b_1, b_2\}$, then the identity $${\rm gcd}(b_1, b_2){\rm lcm}(b_1, b_2) = {b_1}{b_2}$$ proves that the existence of the gcd is equivalent to the existence of the lcm. To treat the case with more than two elements, I first established the following identities by mathematical induction $${d_n}:={\rm gcd}(b_1,\cdots ,b_n) = {\rm gcd}({\rm gcd}(b_1,\cdots ,b_{n-1}), b_n),$$ and $${m_n}:={\rm lcm}(b_1,\cdots ,b_n) = {\rm lcm}({\rm lcm}(b_1,\cdots ,b_{n-1}), b_n).$$ Then assume (as our induction hypothesis) that the existence of the gcd is equivalent to the existence the lcm for any subset consisting of $k$ elements for any $2\leq k\leq n$, and assume that $B$ is an arbitrary subset of $R$ consisting of $n+1$ elements. If $d_{n+1}$ exists, since $$d_{n+1}={\rm gcd}(d_n,b_{n+1}),$$ the existence of $d_n$ implies the existence of $m_n$, (by our induction hypothesis), and by the result for $n =2$, we conclude that $${\rm lcm}(m_n, b_{n+1}) = {\rm lcm}(b_1,\cdots ,b_{n+1})=m_{n+1}$$ must exist. The converse implication is proved with a very similar argument, and the assertion follows by mathematical induction.

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