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Main Question or Discussion Point
In Chapter 1 of his book: "Modules and Rings", John Dauns (on page 7) considers a subset ##T## of an Rmodule ##M## and defines the Rsubmodule generated by ##T## ... for which he uses the notation ##\langle T \rangle## ... ... as follows:
Now, note that Dauns (in Section 12.5) also defines ##\sum M_i = \langle \cup M_i \rangle## ... and so it follows (I think) that if the family of submodules, ##\{ M_i \}_I## spans or generates ##M## ... then we have
##\{ M_i \}_I## generates/spans ##M \Longrightarrow M = \sum M_i = \langle \cup M_i \rangle## ... ... ... ... ... (1)
Note that on page 8, under the heading Observations, Dauns states:
" ... ... if ##1 \in R, \langle T \rangle = \sum \{ tR \mid t \in T \}## ... ... ... ... ... (2)
Now, we have that
(1) (2) ##\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 Rmodule M is said to be generated by a set ##\{ M_\alpha \}_\Delta## of Rmodules if there is an epimorphism ##\bigoplus_\Delta M_\alpha \to M##. ... ... "
The complete Definition 4.1.2 by Bland reads as follows:
Can 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 12.4 to 12.8 ... as follows ...
Now, note that Dauns (in Section 12.5) also defines ##\sum M_i = \langle \cup M_i \rangle## ... and so it follows (I think) that if the family of submodules, ##\{ M_i \}_I## spans or generates ##M## ... then we have
##\{ M_i \}_I## generates/spans ##M \Longrightarrow M = \sum M_i = \langle \cup M_i \rangle## ... ... ... ... ... (1)
Note that on page 8, under the heading Observations, Dauns states:
" ... ... if ##1 \in R, \langle T \rangle = \sum \{ tR \mid t \in T \}## ... ... ... ... ... (2)
Now, we have that
(1) (2) ##\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 Rmodule M is said to be generated by a set ##\{ M_\alpha \}_\Delta## of Rmodules if there is an epimorphism ##\bigoplus_\Delta M_\alpha \to M##. ... ... "
The complete Definition 4.1.2 by Bland reads as follows:
Can 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 12.4 to 12.8 ... as follows ...
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