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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 $$d$$ and $$d'$$ of $$a$$ and $$b$$ coincide: $$(d') = (d)$$. ... ..."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
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 $$d$$ and $$d'$$ of $$a$$ and $$b$$ coincide: $$(d') = (d)$$. ... ..."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