MHB Principal Ideal Domains .... ....

[Math Amateur]

I am reading The Basics of Abstract Algebra by Paul E. Bland ...

I am focused on Section 7.2 Euclidean Domains, Principal Ideal Domains, Unique Factorization Domains ... ...

I need help with the proof of Lemma 7.2.13 ... ... Lemma 7.2.13 reads as follows:https://www.physicsforums.com/attachments/8255Bland states Lemma 7.2.13 but does not prove it ... can someone please demonstrate a proof ...Peter

 

[castor28]

Hi Peter,

If $(a)=(b)$ then $a\in(b)$ and $a=rb$ from some $r\in D$; a similar argument shows that $b=sa$ for some $s\in D$.
We have therefore $a = rb = (rs)a$ and $a(rs-1)=0$. As we are in an integral domain, this implies $rs=1$, which means that $r$ and $s$ are units and $a$ and $b$ are associates.

Conversely, if $a=rb$, we have $a\in(b)$ and $(a)\subset(b)$; a similar argument shows that $(b)\subset(a)$, and therefore $(a)=(b)$.

 

[Math Amateur]

Hi Peter,

If $(a)=(b)$ then $a\in(b)$ and $a=rb$ from some $r\in D$; a similar argument shows that $b=sa$ for some $s\in D$.
We have therefore $a = rb = (rs)a$ and $a(rs-1)=0$. As we are in an integral domain, this implies $rs=1$, which means that $r$ and $s$ are units and $a$ and $b$ are associates.

Conversely, if $a=rb$, we have $a\in(b)$ and $(a)\subset(b)$; a similar argument shows that $(b)\subset(a)$, and therefore $(a)=(b)$.

Thanks castor28 ... appreciate your help ...

Peter