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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