elias001
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- TL;DR Summary
- 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:
Along with the following notations in the screenshot pages:
In the next page (page 4), in Proposition 4.35 it states:
and on page 5 in Proposition 4.38 where it states:
Also in the proof of Proposition 4.38: where in the notation, it says:
In the last page, we also have the
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.
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:
andThe 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 }##
Also on on page 3 in Proposition 4.33 it states: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 }##
##\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:
and##m=\text{ker(coker f)}##
##e=\text{coker(ker f)}##
Also in the proof of Proposition 4.38: where in the notation, it says:
and##\text{(coker f)}f=0##
##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: