Split Sequences - D&F Ch 10 Section 10.5

[Math Amateur]

I am reading Dummit and Foote, CH 10 Section 10.5, Exact Sequences - Projective, Injective and Flat Modules.

As they introduce split sequences, D&F write the following:View attachment 2462

I am concerned at the following statement:

"In this case the module \(\displaystyle B \) contains a sub-module \(\displaystyle C'\) isomorphic to \(\displaystyle C\) (namely \(\displaystyle C' = 0 \oplus C\)) as well as the submodule A, and this submodule complement to A "splits" B into a direct sum ... ... "

Maybe I am being pedantic or I am just confused but it does not seem to me that B contains A as a sub-module or that C' is complement to A (regarding A as an abelian group).

It seems to me that if \(\displaystyle B = A \oplus B \) then certainly B contains both \(\displaystyle A' = A \oplus 0 \) (and NOT A) and \(\displaystyle C' = 0 \oplus C\).

To check that the submodule C' is complement to A' we note that

(1) \(\displaystyle B = A' + C' \)

since

\(\displaystyle A' + C' = \{ (a,0) + (0,c) = (a,c) | a \in A, c \in C \} \)

and we also note that

(2) \(\displaystyle A' \cap C' = (0,0) \)

Given (1) and (2), the submodule \(\displaystyle C'\) is complement to the submodule \(\displaystyle A'\)

Can someone please indicate whether my analysis is correct, and also comment of the D&F text? Should D&F be talking about \(\displaystyle A'\) being a submodule and specifying \(\displaystyle C'\) as complement to \(\displaystyle A'\) (not \(\displaystyle A\))?

Peter

 

[Deveno]

Well, yes, you're "kinda right".

STRICTLY speaking, $A$ is not a sub-module of $A\oplus C$. However, $A$ and $A\oplus 0$ are clearly isomorphic.

Remember, isomorphism is an EQUIVALENCE relation among modules, so we are just replacing "equal" with "the same for all practical (algebraic) purposes".

For example, the real $x$-axis, is a "different thing" than the real line itself (one is PART of a plane, and the other is a line "in it's own little universe"). But in any significant mathematical way, they are "very similar".

In much the same way, a polynomial (a finite power series) is not the same thing as the infinite power series that has the same non-zero coefficients as the polynomial, but padding a finite sequence "with infinite 0's at the end" doesn't change how it BEHAVES much.

There are two different ways to look at mathematical entities (I have touched on this before):

1) As "concrete things", for example $S_3$ could be viewed literally as "possible seating chart re-arrangements of 3 people" (or "bead colorings", an example that is often used to illustrate some group theorems).

2) As (unspecified) entities that merely possesses some desirable PROPERTIES.

With objects of type 1, the objects' ontological status is taken as GIVEN, and properties it has are deduced and verified.

With objects of type 2, consequences of its defining properties are deduced, but it does not follow that some "actual" object has this property.

In other words, the BEING (existence) of objects of type 1 takes precedence (some mathematical Platonists would argue that ALL "true" mathematical objects belong in this category), and the ACTIONS of these objects are secondary. For type 2, the opposite is true, and in "abstract" presentations the defining properties come FIRST, and then EXAMPLES (when they exist) are of secondary importance.

The difference between these two notions are the distinction between "the fourth roots of unity" (that is, the complex numbers $\{1,-1,i,-i\}$, which can be constructed "from the ground up" based on a notion of "existent numbers" (things we count), and "a cyclic group of order 4".

************

What actually happens is a sort of "hybrid", we observe the world, we notice patterns. We abstract from these patterns (a form of inductive reasoning), and then "distill" the patterns into a "minimal form" (deductive reasoning). We then APPLY the distilled patterns back TO the world we observe, for repeated rounds of iteration.

Mathematical "identity" (that is, what mathematical objects actually ARE) is a sort of "mutable quality". A lot depends on CONTEXT, how general or specific our FOCUS is. A mathematician may see a differential equation as "a game to play". A physicist may see it as "a question she needs to answer".

It's good to have some idea of how things work "at both ends of the spectrum". Some people see math as an art form, some see it as a tool to do work with, most are somewhere in-between.

************

In direct response to your question: yes there is a distinction to be made. How important is this distinction?

 

[Math Amateur]

Well, yes, you're "kinda right".

STRICTLY speaking, $A$ is not a sub-module of $A\oplus C$. However, $A$ and $A\oplus 0$ are clearly isomorphic.

Remember, isomorphism is an EQUIVALENCE relation among modules, so we are just replacing "equal" with "the same for all practical (algebraic) purposes".

For example, the real $x$-axis, is a "different thing" than the real line itself (one is PART of a plane, and the other is a line "in it's own little universe"). But in any significant mathematical way, they are "very similar".

In much the same way, a polynomial (a finite power series) is not the same thing as the infinite power series that has the same non-zero coefficients as the polynomial, but padding a finite sequence "with infinite 0's at the end" doesn't change how it BEHAVES much.

There are two different ways to look at mathematical entities (I have touched on this before):

1) As "concrete things", for example $S_3$ could be viewed literally as "possible seating chart re-arrangements of 3 people" (or "bead colorings", an example that is often used to illustrate some group theorems).

2) As (unspecified) entities that merely possesses some desirable PROPERTIES.

With objects of type 1, the objects' ontological status is taken as GIVEN, and properties it has are deduced and verified.

With objects of type 2, consequences of its defining properties are deduced, but it does not follow that some "actual" object has this property.

In other words, the BEING (existence) of objects of type 1 takes precedence (some mathematical Platonists would argue that ALL "true" mathematical objects belong in this category), and the ACTIONS of these objects are secondary. For type 2, the opposite is true, and in "abstract" presentations the defining properties come FIRST, and then EXAMPLES (when they exist) are of secondary importance.

The difference between these two notions are the distinction between "the fourth roots of unity" (that is, the complex numbers $\{1,-1,i,-i\}$, which can be constructed "from the ground up" based on a notion of "existent numbers" (things we count), and "a cyclic group of order 4".

************

What actually happens is a sort of "hybrid", we observe the world, we notice patterns. We abstract from these patterns (a form of inductive reasoning), and then "distill" the patterns into a "minimal form" (deductive reasoning). We then APPLY the distilled patterns back TO the world we observe, for repeated rounds of iteration.

Mathematical "identity" (that is, what mathematical objects actually ARE) is a sort of "mutable quality". A lot depends on CONTEXT, how general or specific our FOCUS is. A mathematician may see a differential equation as "a game to play". A physicist may see it as "a question she needs to answer".

It's good to have some idea of how things work "at both ends of the spectrum". Some people see math as an art form, some see it as a tool to do work with, most are somewhere in-between.

************

In direct response to your question: yes there is a distinction to be made. How important is this distinction?

Thanks Deveno ... definitely answers my questions and allays my anxieties over the issue!

Thanks for the help ...

Peter