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I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some further help in order to fully understand the proof of Proposition 4.3.14 ... ...

Proposition 4.3.14 reads as follows:View attachment 8320

View attachment 8321In the above proof by Bland we read the following:

" ... ... If \(\displaystyle \{ x_1 \}\) is a basis for \(\displaystyle F\), then there is an \(\displaystyle a \in R\) such that \(\displaystyle x = x_1 a\). But \(\displaystyle x\) is primitive, so \(\displaystyle a\) is a unit in \(\displaystyle R\). Hence \(\displaystyle x R = x_1 R\) ... ... "

My question is as follows:

Why in the above quote, does it follow that \(\displaystyle x R = x_1 R\) ... ... ?Is it because \(\displaystyle x = x_1 a\) where \(\displaystyle a\) is a unit ... ... ... ... ... (1)

Hence \(\displaystyle xR = x_1 a R\) ... ... I presume this follows (1)

Therefore \(\displaystyle xR = x_1 (a R )\)

But \(\displaystyle aR = R\) since a is a unit ... ''

So \(\displaystyle xR = x_1 R \)

Is that correct?

Peter