Notation questions about kernel, cokernel, image and coimage

[elias001]

TL;DR

Notational questions on kernel, cokernel, coimage and image in some standard results in beginner level commutative algebra.

The screenshot pages below are taken from the book Advanced Algebra by: Anthony W Knapp, pp. 234-240 I also attached a pdf file of all the screenshots together.

I have notations related questions regarding the yellow hightlighted portions in each page in the below screenshots. I know they seem to be a lot, but i promise they are all related. I included page 1 and 7 for completion and for context continuity of the texts for all the seven pages.

Page 1

Page 2

Page 3

Page 4

Page 5

Page 6

Page 7

In each of the above screenshot pages, in the hightlighted portions are the notations where my questions are derived from.

For pages 2 and pages 3:

The brief form of the definition of kernel is that $u\circ (\text{ker } u)=0$ and $ui'=0\text{ implies }i'=(\text{ker } u) \circ \varphi\text{ uniquely }$

and

The brief form of the definition of cokernel is that $(\text{coker } u)\circ u =0$ and $p'u=0\text{ implies }p'=\psi\circ (\text{coker } u) \text{ uniquely }$

Also on on page 3 in Proposition 4.33 it states:

$\text{ker}(mu)text[ker ]u\text{ and }\text{coker}(ue)=\text{coker }u.$

Along with the following notations in the screenshot pages:

In the next page (page 4), in Proposition 4.35 it states:

$\text{ker(coker(ker u))}=\text{ker }u.$

$\text{coker(ker(coker )}u=\text{coker }u$

and on page 5 in Proposition 4.38 where it states:

$m=\text{ker(coker f)}$

and

$e=\text{coker(ker f)}$

Also in the proof of Proposition 4.38: where in the notation, it says:

$\text{(coker f)}f=0$

and

$f=me=m\text{(ker )}e'=m'e',\text{ where }m'=m\text{ ker }r$

In the last page, we also have the

$\text{coimage f}$

If I am given a homomorphism/mapping/linear transformation $f$ in the context of groups, commutative rings or modules, where $f$ maps from $A$ to $B$. We can define the following:

$\text{ker }f=\{x\in A\mid f(x)=0\},$

$\text{im }f=\{f(x)\in B\mid x\in A\}=f(A),$

$\text{coim }f:=A/\text{ker }f,$

$\text{coker }f:=B/\text{im }f,$

along with $\text{im }f\cong A/\text{ker }a=\text{coim }f$

Then I would often see the two following phrases:

The image of $f$ is the kernel of the cokernel of $f \quad(*)$,

The coimage of $f$ is the cokernel of the kernel of $f\quad(**)$

In math notations, it would be the same as the hightlighted notation on page 5

Also for $(*)$ in math notation, it would be $\text{im }f=\text{ker(coker f)}$ and for $(**)$, in math notation, it would be $\text{coim }f=\text{coker(ker f)}$

It is also well know that $\text{im }f\cong A/\text{ker }f=\text{coim }f$

My questions are as follows:


1. How do I show $\text{im }f=\text{ker(coker }f)=\text{ker }(B/\text{im }f)=f(A), \text{coim }f=\text{coker(ker }f)=\text{coker(B}/\text{im ker }f))=A/\text{ker }f?$


2. In page 6 screenshot, the notations: $\text{(coker }f)f=0, m\text{(ker )}e', m\text{ ker }r$;
for $\text{(coker }f)f$, does it mean $\text{(coker }f)\circ f$, for $m\text{(ker }r)e',$ does it mean $m\circ \text{(ker )}r\circ e'$ and for $m\text{ ker }r,$ does it mean $m\circ \text{ ker }r$? If not, then do the notation mean operationally, like, how do I multiply $f$ and $\text{(coker }f)$, similarly, how does one multiply $m,$, for $m\text{(ker }r)e',$ how do I multiply together $m, \text{(ker }r), e'$, and siimilarly for $m\text{ ker }r?$

Thank you in advance.

 

[martinbn]

You have the map $B\rightarrow B/im(f)$. What is the kernal?

 

[elias001]

@martinbn sorry for my late reply, I just got up and saw your response. Let $p$ denote your map: $p:B\to B/\text{im}(f)$, then we can define the kernel of $p$ as $\text{ker }p=\{y\in B\mid p(y)=0+f(A)=f(A)\},$ specificially, we want $p(y)=y+f(A)=0+f(A)=f(A).$

By the way, I know kernel and cokernel of a map can be talked about respectively in terms of pull backs and push outs using basic category language. I also know that amongst the two concepts: coimage, cokernel. One of these two can't be discussed in either groups, rings or modules. I forgot which ones specifically. But for now, I am trying to stick to concrete contexts like commutative rings, groups or vector spaces, matrices and linear transformations.

 

[martinbn]

Ok, then it seems that it is clear to you that the kernel of the cokernel is the image. So what is your question then!?

 

[elias001]

@martinbn how is my answer show that the kernel of the cokernel is the image. Also, for my question 2. in my post, I still don't know what the notations: $\text{(coker f)}f=0, m\text{(ker )}e', m\text{ ker }r$ mean? Can you also explain that please? Is really confusing me. Thank you in advance.

 

[elias001]

@martinbn @fresh_42 After doing more research, here is what I am still confused about:

If we have a map $f$ that maps from $A$ to $B$ and $f$ is a homomorphism, then the definition of cokernel and coimage are respectively $\frac{B}{\text{im }f},\frac{A}{\text{ker }f}$. We have the usual definition for kernel and image. Both $\frac{B}{\text{im }f},\frac{A}{\text{ker }f}$ are sets; and if we want to discuss properties of $f$ like surjectivity, injectivity, monomorphism, epimorphism using exact sequences, or in the context of additive category; why is it we can consider maps of the form $p:B\to \frac{B}{\text{im }f}$, which is the cokernel of $f$, and similarly $k:\frac{A}{\text{ker }f}\to A$ as the kernel of $f$? We end up having exact sequence of the following form: $\text{ker }f\xrightarrow{\text{ker }f} A\xrightarrow{f} B\xrightarrow{\text{coker }f} \text{coker }f,$, then one gets notation like $f\circ\text{ker }f=0,$ and $\text{coker }f\circ f=0,$ and also, $\text{im }f=\text{coker }(\text{ker }f)=\text{coker }(\text{ker }f\to A)$ and $\text{coim }f=\text{ker }(\text{coker }f)=\text{ker }(B\to \text{coker }f).$

So when discussing cokernel, coimage of a map f, how does one know whether it is in terms of a map or a set?

 

[martinbn]

So when discussing cokernel, coimage of a map f, how does one know whether it is in terms of a map or a set?

From the context. The same way you know what A is. Is it a ring or a vertex of a triangle?

 

[elias001]

@martinbn Can you elaborate on that more, in the of examples of how you would write it in both circumstances. I just want to see examples of how it is stated properly for someone who knows these things better than myself.