Why Does \( a_1 \mid b_1 b_2 \cdots b_n \) in Theorem 7.2.20?

[Math Amateur]

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

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

I need help with the proof of Theorem 7.2.20 ... ... Theorem 7.2.20 and its proof reads as follows:https://www.physicsforums.com/attachments/8280
View attachment 8281
In the last paragraph of the above proof by Bland we read the following:

" ... ... If \(\displaystyle a = a_1 a_2 \ ... \ ... \ a_m = b_1 b_2 \ ... \ ... \ b_n\) where each \(\displaystyle a_i\) and \(\displaystyle b_i\) is irreducible, then \(\displaystyle a_1 \mid b_1 b_2 \ ... \ ... \ b_n\) ... ... "
Can someone please explain exactly and in detail why/how \(\displaystyle a_1 \mid b_1 b_2 \ ... \ ... \ b_n\) ... ... Peter

 

[steenis]

Notate $x=b_1 b_2 \cdots b_n$
What does it mean if $a_1$ divides $x$ ?
It means that there is a $y \in D$ such that $a_1y=x$
Can you tell now what is $y$ ?

 

[Math Amateur]

Notate $x=b_1 b_2 \cdots b_n$
What does it mean if $a_1$ divides $x$ ?
It means that there is a $y \in D$ such that $a_1y=x$
Can you tell now what is $y$ ?

Thanks for the help Steenis ...

Basically you have pointed out that:

\(\displaystyle a_1 ( a_2 a_3 \ ... \ ... \ a_m ) = b_1 b_2 \ ... \ ... \ b_n \)

\(\displaystyle \Longrightarrow a_1 \mid b_1 b_2 \ ... \ ... \ b_n\)The above implies that in what you have written we have \(\displaystyle y = a_2 a_3 \ ... \ ... \ a_m\) ... ...Is that correct?

Peter

 

[steenis]

Correct, and do you understand that therefore $a_1|b_1 \cdots b_n$ ?