# External Direct Sums and the Sum of a Family of Mappings ...

Gold Member
I have an issue/problem that relates to Bland initial treatment of external direct sums including Proposition 2.1.5 ... especially Bland's definition of the sum of a family of mappings ...

Bland's text on this is as follows:

In the above text by Bland we read the following:

" ... ... We now need the concept of a family of mappings. If ##f_\alpha \ : \ M_\alpha \rightarrow N## is an R-linear mapping for each ##\alpha \in \Delta##, where ##N## is a fixed R-module, then ##f \ : \ \bigoplus_\Delta M_\alpha \rightarrow N## defined by ##f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x)## ... ... "

But in the last sentence of the proof of Proposition 2.1.5 ( ... again, see above text by Bland ... ) we read:

" ... ... If ##( x_\alpha ) \in \bigoplus_\Delta M_\alpha##, then ##f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x_\alpha)## ... ... "

So ... in the text above the Proposition we have ... ##f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x)## ... ... and in the proof of the proposition we have ##f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x_\alpha)## ...

... which of these is correct ... or in some strange way, are they both correct ...

... I note that x is mentioned in the definition of the canonical injections above ..

Can someone please clarify ... ?

But ... if ##f## is defined by ##f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x_\alpha)## ... then I have a further problem ...

... we know that ##f \ : \ M_\alpha \rightarrow N## ... that is the domain of ##f_\alpha## is ##M_\alpha## ... BUT ...PROBLEM ... ##( x_\alpha ) \in \bigoplus_\Delta M_\alpha## and ##( x_\alpha ) \notin M_\alpha## ...

... can someone please clarify ?

Hope someone can help ...

Peter

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andrewkirk
Homework Helper
Gold Member
The second one is correct. I think the ##\alpha## subscript was left off the ##x## in the first one by mistake.

Re the further problem. You say we know that ##f:M_\alpha\to N##. I can't see that the author has said that. Right now, because of an internet problem, I can only see your latex, not the png from the text. But in the latex you have quoted the author as writing that the domain of ##f## is ##\bigoplus_\Delta M_\alpha##, not ##M_\alpha## which is the domain of ##f_\alpha## (note the subscript on the ##f##). As long as we stick to that, I think we'll be OK.

Gold Member
The second one is correct. I think the ##\alpha## subscript was left off the ##x## in the first one by mistake.

Re the further problem. You say we know that ##f:M_\alpha\to N##. I can't see that the author has said that. Right now, because of an internet problem, I can only see your latex, not the png from the text. But in the latex you have quoted the author as writing that the domain of ##f## is ##\bigoplus_\Delta M_\alpha##, not ##M_\alpha## which is the domain of ##f_\alpha## (note the subscript on the ##f##). As long as we stick to that, I think we'll be OK.

Sorry Andrew ... it was a typo ...

I wrote:

" ... ... ... we know that ##f \ : \ M_\alpha \rightarrow N## ... that is the domain of ##f_\alpha## is ##M_\alpha## ... "

but I meant

" ... ... ##f_\alpha \ : \ M_\alpha \rightarrow N## ... that is the domain of ##f_\alpha## is ##M_\alpha## ... ... "

Peter

andrewkirk
Homework Helper
Gold Member
Internet is mended now so I can see the png. Is your problem solved? If not can you elaborate on what the remaining difficulty is?

Math Amateur
Gold Member
Thanks Andrew ... Issues are resolved ...

Peter