I Notation questions about kernel, cokernel, image and coimage

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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

Anthony W. Knapp-Advanced Algebra (2016) pp 264- 270-images-0.webp


Page 2

Anthony W. Knapp-Advanced Algebra (2016) pp 264- 270-images-1.webp


Page 3

Anthony W. Knapp-Advanced Algebra (2016) pp 264- 270-images-2.webp


Page 4

Anthony W. Knapp-Advanced Algebra (2016) pp 264- 270-images-3.webp


Page 5

Anthony W. Knapp-Advanced Algebra (2016) pp 264- 270-images-4.webp


Page 6

Anthony W. Knapp-Advanced Algebra (2016) pp 264- 270-images-5.webp


Page 7

Anthony W. Knapp-Advanced Algebra (2016) pp 264- 270-images-6.webp



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.
 
Last edited:
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You have the map ##B\rightarrow B/im(f)##. What is the kernal?
 
@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.
 
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!?
 
@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.
 
@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?
 
elias001 said:
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?
 
@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.
 
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