Re: Principal ideal domain

Quote:

Originally Posted by

**jishnuray** R is an integral domain.Then prove that if the following two conditions hold then R is a principal ideal domain(PID)

1)any two non zero elements a, b in R have a greatest common divisor which can be written in the form ra+sb for some r,s belonging to R.

2)['a_1 denotes a suffix 1]

If a_1,a_2,a_3,...... are non zero elements of R such that a_(i+1) divides a_i for all i then there is a positive integer N such that a_n is unit times a_N for all n greater than or equal to N.

How to do the sum?

let $\displaystyle I$ be any non-zero ideal of $\displaystyle R$ and define $\displaystyle S = \{Ra : \ a \in I \}$. use condition 2) in your problem to show that $\displaystyle S$ has a maximal element, say $\displaystyle Rb$. the claim is that $\displaystyle I = Rb$. suppose our claim is not true and choose $\displaystyle a \in I \setminus Rb$. then by condition 1) we have $\displaystyle Ra+Rb=Rd$, where $\displaystyle d = \gcd(a,b)$. clearly $\displaystyle d \in I$ because $\displaystyle a,b \in I$. thus $\displaystyle Rd \in S$. we also have $\displaystyle a \in Rd \setminus Rb$ and so $\displaystyle Rb \subset Rd$, contradicting the maximality of $\displaystyle Rb$ in $\displaystyle S$. Q.E.D.

Re: Principal ideal domain

can this be proved without the use of Zorn's lemma? perhaps i am not understanding the question, but condition two looks like an ascending chain condition on the principal ideals:

$\displaystyle (a_i)$.

i was thinking about this, and i ran into the following difficulty:

given any (non-zero) ideal I, we can form a chain of ideals by taking:

$\displaystyle I_1 = (a_1)$ where $\displaystyle a_1 \in I\setminus{0}$.

$\displaystyle I_2 = (a_2)$ where $\displaystyle a_2$ is chosen by finding $\displaystyle r_1 \in I\setminus I_1$ and setting $\displaystyle a_2 = gcd(r_1,a_1)$, so that $\displaystyle a_2|a_1$,

$\displaystyle I_3 = (a_3)$ where $\displaystyle a_3$ is chosen by finding $\displaystyle r_2 \in I\setminus I_2$ and setting $\displaystyle a_3 = gcd(r_2,a_2)$, so that $\displaystyle a_3|a_2$,

and so on, which by (2) terminates with some $\displaystyle I_N =(a_N)$.

now, it would be nice if we could conclude that $\displaystyle I = I_N$, but perhaps for some other choice of the $\displaystyle \{a_i\}$ we would wind up with "a different N". that is, how do we know that we can find an N that makes every element in $\displaystyle I\setminus I_N$ an associate of $\displaystyle a_N$? (i think this is where Zorn's lemma comes in. if i understand what you wrote, each Ra in your answer, is one of my "$\displaystyle I_N$"s).

my apologies if i have stated this badly.

Re: Principal ideal domain

Quote:

Originally Posted by

**Deveno** can this be proved without the use of Zorn's lemma? perhaps i am not understanding the question, but condition two looks like an ascending chain condition on the principal ideals:

$\displaystyle (a_i)$.

i was thinking about this, and i ran into the following difficulty:

given any (non-zero) ideal I, we can form a chain of ideals by taking:

$\displaystyle I_1 = (a_1)$ where $\displaystyle a_1 \in I\setminus{0}$.

$\displaystyle I_2 = (a_2)$ where $\displaystyle a_2$ is chosen by finding $\displaystyle r_1 \in I\setminus I_1$ and setting $\displaystyle a_2 = gcd(r_1,a_1)$, so that $\displaystyle a_2|a_1$,

$\displaystyle I_3 = (a_3)$ where $\displaystyle a_3$ is chosen by finding $\displaystyle r_2 \in I\setminus I_2$ and setting $\displaystyle a_3 = gcd(r_2,a_2)$, so that $\displaystyle a_3|a_2$,

and so on, which by (2) terminates with some $\displaystyle I_N =(a_N)$.

now, it would be nice if we could conclude that $\displaystyle I = I_N$, but perhaps for some other choice of the $\displaystyle \{a_i\}$ we would wind up with "a different N". that is, how do we know that we can find an N that makes every element in $\displaystyle I\setminus I_N$ an associate of $\displaystyle a_N$? (i think this is where Zorn's lemma comes in. if i understand what you wrote, each Ra in your answer, is one of my "$\displaystyle I_N$"s).

my apologies if i have stated this badly.

yes, you're right. we don't need Zorn's lemma to show that $\displaystyle S$ has a maximal element. i removed it from my proof.

suppose that $\displaystyle S$ has no maximal element and choose $\displaystyle a_1 \in I$. since$\displaystyle Ra_1$ is not maximal in $\displaystyle S$, we may choose $\displaystyle a_2 \in I$ such that $\displaystyle Ra_1 \subset Ra_2$. we continue this way to find an ascending chain of elements of $\displaystyle S$ which does not stop, contradicting condition 2).