This is from Page 101 of Rotman's book on homological algebra. I am not sure for the proof of ii, why there are as many generators of P as basis elements? Any help would be appreciated! Thank you!
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1The assumption here is $P$ is finitely generated. So, it has $n$ generators for some $n>0$. – jijijojo Jun 21 '20 at 15:16
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Hi jijijojo, sorry I didn't make the question clear. I meant why the number of generators is the same as the number of the basis elements. Thx very much! – scsnm Jun 22 '20 at 03:43
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It’s the other way around: the number of elements in the basis is the same as the number of generators for $P$. It’s the number of generators of $P$ that determines the size of the basis, not the other way around. – Arturo Magidin Jun 22 '20 at 03:59
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P.S. Please don’t use images; they are not searchable, they may not be displayed properly, and they are not accessible to people who use screen readers. – Arturo Magidin Jun 22 '20 at 04:00