 #1
Math Amateur
Gold Member
 1,067
 47
I am reading T. S. Blyth's book "Module Theory: An Approach to Linear Algebra" ... ... and am currently focussed on Chapter 1: Modules, Vector Spaces and Algebras ... ...
I need help with a basic and possibly simple aspect of Theorem 2.3 ...
Since the answer to my question may depend on Blyth's previous definitions and theorems I am providing some relevant text from Blyth prior to Theorem 2.3 ... but those confident with the theory obviously can go straight to the theorem at the bottom of the scanned text ...
Theorem 2.3 together with some relevant prior definitions and theorems reads as follows: (Theorem 2,3 at end of text fragment)
In the above text (near the end) we read, in the statement of Theorem 2.3:
" ... ... then the submodule generated by ##\bigcup_{ i \in I } M_i## consists of all finite sums of the form ##\sum_{ j \in J } m_j## ... ... "
The above statement seems to assume we take one element from each ##M_j## in forming the sum ##\sum_{ j \in J } m_j## ... ... but how do we know a linear combination does not take more than one element from a particular ##M_j##, say ##M_{ j_0 }## ... ... or indeed all elements from one particular ##M_j## ... rather than one element from each submodule in the family ##\{ M_i \}_{ i \in I}## ...
Hope someone can clarify this ...
Peter
I need help with a basic and possibly simple aspect of Theorem 2.3 ...
Since the answer to my question may depend on Blyth's previous definitions and theorems I am providing some relevant text from Blyth prior to Theorem 2.3 ... but those confident with the theory obviously can go straight to the theorem at the bottom of the scanned text ...
Theorem 2.3 together with some relevant prior definitions and theorems reads as follows: (Theorem 2,3 at end of text fragment)
In the above text (near the end) we read, in the statement of Theorem 2.3:
" ... ... then the submodule generated by ##\bigcup_{ i \in I } M_i## consists of all finite sums of the form ##\sum_{ j \in J } m_j## ... ... "
The above statement seems to assume we take one element from each ##M_j## in forming the sum ##\sum_{ j \in J } m_j## ... ... but how do we know a linear combination does not take more than one element from a particular ##M_j##, say ##M_{ j_0 }## ... ... or indeed all elements from one particular ##M_j## ... rather than one element from each submodule in the family ##\{ M_i \}_{ i \in I}## ...
Hope someone can clarify this ...
Peter
Attachments

121 KB Views: 642

98 KB Views: 506

142.1 KB Views: 702