Principal Ideal Rings and GCDs .... .... Bland Proposition 4.3.3

[Math Amateur]

I am reading Paul E. Bland's book, "Rings and Their Modules".

I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help to fully understand the proof of part of Proposition 4.3.3 ... ...

Proposition 4.3.3 reads as follows:View attachment 8247
View attachment 8248
In the above proof by Bland we read the following:

"... ... If \(\displaystyle (d) = a_1 R + a_2 R + \ ... \ ... \ + a_n R\), then each \(\displaystyle a_i\) is in \(\displaystyle (d)\) ... ... "Can someone please explain how \(\displaystyle (d) = a_1 R + a_2 R + \ ... \ ... \ + a_n R\) implies each \(\displaystyle a_i\) is in \(\displaystyle (d)\) ... ..Peter

 

[Euge]

Hi Peter,
$$a_1 = a_1\cdot 1 + a_2 \cdot 0 + \cdots + a_n \cdot 0 \in (d)$$ and similarly for the other $a_i$.