what is the cokernel of this map? $ ain R, f_a:Mrightarrow M f_a(m)=acdot m$ Announcing...
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what is the cokernel of this map? $ ain R, f_a:Mrightarrow M f_a(m)=acdot m$
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Isomorphism modulo torsion is stable under tensor productsA question about exact sequences of module homomorphisms.If $mu_r(m)=rm$ then ker $(mu_r)$ and coker $(mu_r)$ are modules over $R/I$about homomorphismes of modulesFor an exact sequence $0to M_1overset{f_1}tocdotsoverset{f_r}to M_rto0$ is it true that $l(M_i)-l(M_{i+1})=l(ker(f_i))-l(ker(f_{i+1}))$?Let $mathscr{A}$ be a chain complex. Show that the kernel of the map $A_n/B_n rightarrow Z_{n-1} $ is isomorphic to $H_{n}(A)$.Kernel of a natural map is a direct summand of the covariant extensionMonomorphisms in abelian categories: $f=ker(text{coker} f)$Does a map between the localizations have to be induced by something?Understandin kernel, image and cokernel of a map
$begingroup$
consider a commutative unitary ring $R$ and $M$ is an $R$-module.
For $ ain R, f_a:Mrightarrow M f_a(m)=acdot m$ is a homomorphism.
What is the kernel and cokernel of this map?
The kernel I found is $ker={ min M | f_a(m)=O_m}= { min M | acdot m=O_m}={O_m} $
I am not sure about the final equality.
I know that cokernel is equal to $frac{M}{Imf_a}$
how can I find cokernel?
how can I express cokernel of any map as a set? like for kernel $ker={ min M | f_a(m)=O_m}$ how can I write cokernel?
I mean
$coker={ m+Imf_a | ? }$
please help
abstract-algebra modules
asked Mar 24 at 20:37
HitmanHitman
1939
$endgroup$
|
show 3 more comments
$begingroup$
consider a commutative unitary ring $R$ and $M$ is an $R$-module.
For $ ain R, f_a:Mrightarrow M f_a(m)=acdot m$ is a homomorphism.
What is the kernel and cokernel of this map?
The kernel I found is $ker={ min M | f_a(m)=O_m}= { min M | acdot m=O_m}={O_m} $
I am not sure about the final equality.
I know that cokernel is equal to $frac{M}{Imf_a}$
how can I find cokernel?
how can I express cokernel of any map as a set? like for kernel $ker={ min M | f_a(m)=O_m}$ how can I write cokernel?
I mean
$coker={ m+Imf_a | ? }$
please help
abstract-algebra modules
asked Mar 24 at 20:37
HitmanHitman
1939
$endgroup$
2
$begingroup$
In your description of the kernel, why is the last equality true?
$endgroup$
– Jason DeVito
Mar 24 at 20:39
$begingroup$
@JasonDeVito I thought for for $aneq0 $ m should be $O_m$
$endgroup$
– Hitman
Mar 24 at 20:43
1
$begingroup$
Consider $M = R = mathbb{Z}/4mathbb{Z}$ with $f_a(m) = am$. Then $operatorname{ker} f_2 neq {0}$. Also, I'm not sure you're going to get a nice description of either the kernel or cokernel in the generality you asked. I am not an algebraist, though....
$endgroup$
– Jason DeVito
Mar 24 at 20:45
$begingroup$
@JasonDeVito so shall I write it as ${ min M | acdot m=O_m}$ ?
$endgroup$
– Hitman
Mar 24 at 20:47
1
$begingroup$
isn't that just $M/aM$ and that should be as good as it gets.
$endgroup$
– Enkidu
Mar 25 at 8:07
|
show 3 more comments
$begingroup$
consider a commutative unitary ring $R$ and $M$ is an $R$-module.
For $ ain R, f_a:Mrightarrow M f_a(m)=acdot m$ is a homomorphism.
What is the kernel and cokernel of this map?
The kernel I found is $ker={ min M | f_a(m)=O_m}= { min M | acdot m=O_m}={O_m} $
I am not sure about the final equality.
I know that cokernel is equal to $frac{M}{Imf_a}$
how can I find cokernel?
how can I express cokernel of any map as a set? like for kernel $ker={ min M | f_a(m)=O_m}$ how can I write cokernel?
I mean
$coker={ m+Imf_a | ? }$
please help
abstract-algebra modules
asked Mar 24 at 20:37
HitmanHitman
1939
$endgroup$
consider a commutative unitary ring $R$ and $M$ is an $R$-module.
For $ ain R, f_a:Mrightarrow M f_a(m)=acdot m$ is a homomorphism.
What is the kernel and cokernel of this map?
The kernel I found is $ker={ min M | f_a(m)=O_m}= { min M | acdot m=O_m}={O_m} $
I am not sure about the final equality.
I know that cokernel is equal to $frac{M}{Imf_a}$
how can I find cokernel?
how can I express cokernel of any map as a set? like for kernel $ker={ min M | f_a(m)=O_m}$ how can I write cokernel?
I mean
$coker={ m+Imf_a | ? }$
please help
abstract-algebra modules
abstract-algebra modules
asked Mar 24 at 20:37
HitmanHitman
1939
asked Mar 24 at 20:37
HitmanHitman
1939
edited Mar 24 at 20:52
Hitman
asked Mar 24 at 20:37
HitmanHitman
1939
asked Mar 24 at 20:37
HitmanHitman
1939
asked Mar 24 at 20:37
HitmanHitman
1939
1939
2
$begingroup$
In your description of the kernel, why is the last equality true?
$endgroup$
– Jason DeVito
Mar 24 at 20:39
$begingroup$
@JasonDeVito I thought for for $aneq0 $ m should be $O_m$
$endgroup$
– Hitman
Mar 24 at 20:43
1
$begingroup$
Consider $M = R = mathbb{Z}/4mathbb{Z}$ with $f_a(m) = am$. Then $operatorname{ker} f_2 neq {0}$. Also, I'm not sure you're going to get a nice description of either the kernel or cokernel in the generality you asked. I am not an algebraist, though....
$endgroup$
– Jason DeVito
Mar 24 at 20:45
$begingroup$
@JasonDeVito so shall I write it as ${ min M | acdot m=O_m}$ ?
$endgroup$
– Hitman
Mar 24 at 20:47
1
$begingroup$
isn't that just $M/aM$ and that should be as good as it gets.
$endgroup$
– Enkidu
Mar 25 at 8:07
|
show 3 more comments
2
$begingroup$
In your description of the kernel, why is the last equality true?
$endgroup$
– Jason DeVito
Mar 24 at 20:39
$begingroup$
@JasonDeVito I thought for for $aneq0 $ m should be $O_m$
$endgroup$
– Hitman
Mar 24 at 20:43
1
$begingroup$
Consider $M = R = mathbb{Z}/4mathbb{Z}$ with $f_a(m) = am$. Then $operatorname{ker} f_2 neq {0}$. Also, I'm not sure you're going to get a nice description of either the kernel or cokernel in the generality you asked. I am not an algebraist, though....
$endgroup$
– Jason DeVito
Mar 24 at 20:45
$begingroup$
@JasonDeVito so shall I write it as ${ min M | acdot m=O_m}$ ?
$endgroup$
– Hitman
Mar 24 at 20:47
1
$begingroup$
isn't that just $M/aM$ and that should be as good as it gets.
$endgroup$
– Enkidu
Mar 25 at 8:07
2
2
$begingroup$
In your description of the kernel, why is the last equality true?
$endgroup$
– Jason DeVito
Mar 24 at 20:39
$begingroup$
In your description of the kernel, why is the last equality true?
$endgroup$
– Jason DeVito
Mar 24 at 20:39
$begingroup$
@JasonDeVito I thought for for $aneq0 $ m should be $O_m$
$endgroup$
– Hitman
Mar 24 at 20:43
$begingroup$
@JasonDeVito I thought for for $aneq0 $ m should be $O_m$
$endgroup$
– Hitman
Mar 24 at 20:43
1
1
$begingroup$
Consider $M = R = mathbb{Z}/4mathbb{Z}$ with $f_a(m) = am$. Then $operatorname{ker} f_2 neq {0}$. Also, I'm not sure you're going to get a nice description of either the kernel or cokernel in the generality you asked. I am not an algebraist, though....
$endgroup$
– Jason DeVito
Mar 24 at 20:45
$begingroup$
Consider $M = R = mathbb{Z}/4mathbb{Z}$ with $f_a(m) = am$. Then $operatorname{ker} f_2 neq {0}$. Also, I'm not sure you're going to get a nice description of either the kernel or cokernel in the generality you asked. I am not an algebraist, though....
$endgroup$
– Jason DeVito
Mar 24 at 20:45
$begingroup$
@JasonDeVito so shall I write it as ${ min M | acdot m=O_m}$ ?
$endgroup$
– Hitman
Mar 24 at 20:47
$begingroup$
@JasonDeVito so shall I write it as ${ min M | acdot m=O_m}$ ?
$endgroup$
– Hitman
Mar 24 at 20:47
1
1
$begingroup$
isn't that just $M/aM$ and that should be as good as it gets.
$endgroup$
– Enkidu
Mar 25 at 8:07
$begingroup$
isn't that just $M/aM$ and that should be as good as it gets.
$endgroup$
– Enkidu
Mar 25 at 8:07
|
show 3 more comments
0
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2
$begingroup$
In your description of the kernel, why is the last equality true?
$endgroup$
– Jason DeVito
Mar 24 at 20:39
$begingroup$
@JasonDeVito I thought for for $aneq0 $ m should be $O_m$
$endgroup$
– Hitman
Mar 24 at 20:43
1
$begingroup$
Consider $M = R = mathbb{Z}/4mathbb{Z}$ with $f_a(m) = am$. Then $operatorname{ker} f_2 neq {0}$. Also, I'm not sure you're going to get a nice description of either the kernel or cokernel in the generality you asked. I am not an algebraist, though....
$endgroup$
– Jason DeVito
Mar 24 at 20:45
$begingroup$
@JasonDeVito so shall I write it as ${ min M | acdot m=O_m}$ ?
$endgroup$
– Hitman
Mar 24 at 20:47
1
$begingroup$
isn't that just $M/aM$ and that should be as good as it gets.
$endgroup$
– Enkidu
Mar 25 at 8:07