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\(\displaystyle \{ M_i \}_I\) generates/spans \(\displaystyle M \Longrightarrow M = \sum M_i = \langle \cup M_i \rangle\) ... ... ... ... ... (1)Note that on page 8, under the heading Observations, Dauns states:

" ... ... if \(\displaystyle 1 \in R, \langle T \rangle = \sum \{ tR \mid t \in T \}\) ... ... ... ... ... (2)Now, we have that

(1) (2) \(\displaystyle \Longrightarrow M = \sum M_i = \langle \cup M_i \rangle = \sum \{ tR \mid t \in \cup M_i \}\) ... ... ... ... ... (3)But ... how do we reconcile Dauns' definitions with Bland's Definition 4.1.2 which states

" ... ... An R-module \(\displaystyle M\) is said to be generated by a set \(\displaystyle \{ M_\alpha \}_\Delta\) of R-modules if there is an epimorphism \(\displaystyle \bigoplus_\Delta M_\alpha \to M\). ... ... "The complete Definition 4.1.2 by Bland reads as follows:View attachment 8152Can someone please explain how to reconcile Dauns' and Bland's definitions ...

Just a note ... I feel that Dauns definition has more the "feel" of something being generated ...

To give readers of the above post the context including the notation of Dauns approach I am providing the text of Sections 1-2.4 to 1-2.8 ... as follows ...

View attachment 8153

View attachment 8154Hope that text helps ...

Peter