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I am focused on Section 4.2: Noetherian and Artinian Modules and need some further help to fully understand the proof of part of Proposition 4.2.7 ... ...

Proposition 4.2.7 reads as follows:https://www.physicsforums.com/attachments/8209In the above proof, when Bland is dealing with the converse, we read the following:

" ... ... Then the short exact sequence

\(\displaystyle 0 \longrightarrow \bigoplus_{ i = 1 }^{ n-1 } M_i \longrightarrow \bigoplus_{ i = 1 }^{ n } M_i \longrightarrow M_n \longrightarrow 0\) ... ... "Now I understand the induction argument that Bland goes on to talk about ... but we need to establish that the sequence of modules given is indeed a short exact sequence ... so ... some questions follow ... ... When an author gives a short exact sequence such as

\(\displaystyle 0 \longrightarrow \bigoplus_{ i = 1 }^{ n-1 } M_i \longrightarrow \bigoplus_{ i = 1 }^{ n } M_i \longrightarrow M_n \longrightarrow 0\)

is he/she implying that the R-linear mappings (R-module homomorphisms ...) involved are obvious ... ?If that is the case then what are the obvious R-module homomorphisms in the case of

\(\displaystyle 0 \longrightarrow \bigoplus_{ i = 1 }^{ n-1 } M_i \stackrel{ f }{ \longrightarrow } \bigoplus_{ i = 1 }^{ n } M_i \stackrel{ g }{ \longrightarrow } M_n \longrightarrow 0\)

and how does \(\displaystyle \text{ I am } f = \text{ Ker } g\) ...Peter