How Do GCDs Coincide in Principal Ideals According to Rotman's Proposition 3.41?

[Math Amateur]

I am reading Joseph J.Rotman's book, A First Course in Abstract Algebra.

I am currently focused on Section 3. Polynomials

I need help with the a statement of Rotman's concerning the definition of a gcd in a general domain ... the definition

The relevant section of Rotman's text reads as follows:View attachment 4544
View attachment 4545
In the above text we read the following:

" ... ... By Proposition 3.41, the principal ideals generated by two gcd's \(\displaystyle d\) and \(\displaystyle d'\) of \(\displaystyle a\) and \(\displaystyle b\) coincide: \(\displaystyle (d') = (d)\). ... ..."Can someone please help me to prove (rigorously and formally) that this statement actually follows from Proposition 3.41?

Hope someone can help ...

Peter
*** NOTE ***

Proposition 3.41 reads as follows:https://www.physicsforums.com/attachments/4546

 

[caffeinemachine]

I am reading Joseph J.Rotman's book, A First Course in Abstract Algebra.

I am currently focused on Section 3. Polynomials

I need help with the a statement of Rotman's concerning the definition of a gcd in a general domain ... the definition

The relevant section of Rotman's text reads as follows:

In the above text we read the following:

" ... ... By Proposition 3.41, the principal ideals generated by two gcd's \(\displaystyle d\) and \(\displaystyle d'\) of \(\displaystyle a\) and \(\displaystyle b\) coincide: \(\displaystyle (d') = (d)\). ... ..."Can someone please help me to prove (rigorously and formally) that this statement actually follows from Proposition 3.41?

Hope someone can help ...

Peter
*** NOTE ***

Proposition 3.41 reads as follows:

I don't see how Prop. 3.41 applies. It's pretty straightforward anyway. If $d$ and $d'$ are gcds of $a$ and $b$, then $d|d'$ and $d'|d$. Thus $(d)\subseteq (d')$ and $(d')\subseteq (d)$.