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characterization of submodule of and of linear application.
Clash Royale CLAN TAG#URR8PPP
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Let $A$ a commutative ring, $G$ an abelian group and $K$ a field.
1) Characterize submodules of $A$ for its natural structure of $A$-module
2) Characterize submodules of $G$ for its natural structure of $mathbb Z$-module
3) Characterize submodules of $K[X]$ for its natural structure of $K[X]$-module.
4) Let $G'$ an abelian group. Characterize linear applications $Gto G'$ for their structure of $mathbb Z$-module.
5) Let $E$ and $E'$ two $K$-vectors spaces, $uin mathcal L(E)$ and $u'in mathcal L(E')$. Characterize linear application $Eto E'$ for the structure of $K[X]$-module associated to $u$ and $u'$.
I'm not really sure what to do.
1) $S$ is a submodule of $A$ if $0in S$, $asin S$ for all $ain A$ and all $sin S$ and $s+s'in S$ for all $s,s'in S$ ? Is it the question ?
2) $H$ is a submodule if $0in H$, $kgin H$ for all $kin mathbb Z$ and all $gin G$ and $h+h'in H$ for all $h,h'in H$ ? Is it the question ?
3) $S[X]$ is a $K[X]$-submodule if $0in S[X]$, $p(x)+q(x)in S[X]$ for all $p(x),q(x)in S[X]$ and $p(x)q(x)in S[X]$ for all $p(X)in K[X]$ and all $q(x)in S[X]$ ? Is it the question ?
4) A linear map $f:Gto G'$ is a map s.t. $f(kg)=kf(g)$ for all $kin mathbb Z$ and all $gin G$ and $f(g+h)=f(g)+f(h)$ for all $g,hin G$ ? Is it the question ?
5) $f:Eto E'$ is linear if $f(p(x) y)=p(x)f(y)$ for all $p(x)in K[X]$ and all $yin E$ and $f(y+y')=f(y)+f(y')$ for all $y,y'in E$ ? Is it the question ?
abstract-algebra modules
edited Aug 2 at 2:59
Michael Hardy
204k23185460
asked Aug 1 at 17:37
MathBeginner
695312
add a comment |Â
up vote
1
down vote
favorite
Let $A$ a commutative ring, $G$ an abelian group and $K$ a field.
1) Characterize submodules of $A$ for its natural structure of $A$-module
2) Characterize submodules of $G$ for its natural structure of $mathbb Z$-module
3) Characterize submodules of $K[X]$ for its natural structure of $K[X]$-module.
4) Let $G'$ an abelian group. Characterize linear applications $Gto G'$ for their structure of $mathbb Z$-module.
5) Let $E$ and $E'$ two $K$-vectors spaces, $uin mathcal L(E)$ and $u'in mathcal L(E')$. Characterize linear application $Eto E'$ for the structure of $K[X]$-module associated to $u$ and $u'$.
I'm not really sure what to do.
1) $S$ is a submodule of $A$ if $0in S$, $asin S$ for all $ain A$ and all $sin S$ and $s+s'in S$ for all $s,s'in S$ ? Is it the question ?
2) $H$ is a submodule if $0in H$, $kgin H$ for all $kin mathbb Z$ and all $gin G$ and $h+h'in H$ for all $h,h'in H$ ? Is it the question ?
3) $S[X]$ is a $K[X]$-submodule if $0in S[X]$, $p(x)+q(x)in S[X]$ for all $p(x),q(x)in S[X]$ and $p(x)q(x)in S[X]$ for all $p(X)in K[X]$ and all $q(x)in S[X]$ ? Is it the question ?
4) A linear map $f:Gto G'$ is a map s.t. $f(kg)=kf(g)$ for all $kin mathbb Z$ and all $gin G$ and $f(g+h)=f(g)+f(h)$ for all $g,hin G$ ? Is it the question ?
5) $f:Eto E'$ is linear if $f(p(x) y)=p(x)f(y)$ for all $p(x)in K[X]$ and all $yin E$ and $f(y+y')=f(y)+f(y')$ for all $y,y'in E$ ? Is it the question ?
abstract-algebra modules
edited Aug 2 at 2:59
Michael Hardy
204k23185460
asked Aug 1 at 17:37
MathBeginner
695312
3
No. I very much doubt the author of these questions intended you to regurgitate definitions as you are doing. In the first two questions, you can give a simple name to what they have described as submodules. Each one has a familiar and simple alternative name. The third one is not much different: you can describe the submodules because of the special structure of the ideals of $K[X]$ (oops, dropped a hint). I can't help with the 5th because I don't exactly understand what is written.
– rschwieb
Aug 1 at 17:53
@rschwieb If $V$ is a vector space over $K$, and $T in L(V)$, then $V$ together with $T$ can be thought of as a $K[X]$-module where $X$ acts on a vector $v$ by $T$. That is, $Xv=T(v)$. The rest of the action is given from this.
– Jonathan Dunay
Aug 1 at 19:51
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $A$ a commutative ring, $G$ an abelian group and $K$ a field.
1) Characterize submodules of $A$ for its natural structure of $A$-module
2) Characterize submodules of $G$ for its natural structure of $mathbb Z$-module
3) Characterize submodules of $K[X]$ for its natural structure of $K[X]$-module.
4) Let $G'$ an abelian group. Characterize linear applications $Gto G'$ for their structure of $mathbb Z$-module.
5) Let $E$ and $E'$ two $K$-vectors spaces, $uin mathcal L(E)$ and $u'in mathcal L(E')$. Characterize linear application $Eto E'$ for the structure of $K[X]$-module associated to $u$ and $u'$.
I'm not really sure what to do.
1) $S$ is a submodule of $A$ if $0in S$, $asin S$ for all $ain A$ and all $sin S$ and $s+s'in S$ for all $s,s'in S$ ? Is it the question ?
2) $H$ is a submodule if $0in H$, $kgin H$ for all $kin mathbb Z$ and all $gin G$ and $h+h'in H$ for all $h,h'in H$ ? Is it the question ?
3) $S[X]$ is a $K[X]$-submodule if $0in S[X]$, $p(x)+q(x)in S[X]$ for all $p(x),q(x)in S[X]$ and $p(x)q(x)in S[X]$ for all $p(X)in K[X]$ and all $q(x)in S[X]$ ? Is it the question ?
4) A linear map $f:Gto G'$ is a map s.t. $f(kg)=kf(g)$ for all $kin mathbb Z$ and all $gin G$ and $f(g+h)=f(g)+f(h)$ for all $g,hin G$ ? Is it the question ?
5) $f:Eto E'$ is linear if $f(p(x) y)=p(x)f(y)$ for all $p(x)in K[X]$ and all $yin E$ and $f(y+y')=f(y)+f(y')$ for all $y,y'in E$ ? Is it the question ?
abstract-algebra modules
edited Aug 2 at 2:59
Michael Hardy
204k23185460
asked Aug 1 at 17:37
MathBeginner
695312
Let $A$ a commutative ring, $G$ an abelian group and $K$ a field.
1) Characterize submodules of $A$ for its natural structure of $A$-module
2) Characterize submodules of $G$ for its natural structure of $mathbb Z$-module
3) Characterize submodules of $K[X]$ for its natural structure of $K[X]$-module.
4) Let $G'$ an abelian group. Characterize linear applications $Gto G'$ for their structure of $mathbb Z$-module.
5) Let $E$ and $E'$ two $K$-vectors spaces, $uin mathcal L(E)$ and $u'in mathcal L(E')$. Characterize linear application $Eto E'$ for the structure of $K[X]$-module associated to $u$ and $u'$.
I'm not really sure what to do.
1) $S$ is a submodule of $A$ if $0in S$, $asin S$ for all $ain A$ and all $sin S$ and $s+s'in S$ for all $s,s'in S$ ? Is it the question ?
2) $H$ is a submodule if $0in H$, $kgin H$ for all $kin mathbb Z$ and all $gin G$ and $h+h'in H$ for all $h,h'in H$ ? Is it the question ?
3) $S[X]$ is a $K[X]$-submodule if $0in S[X]$, $p(x)+q(x)in S[X]$ for all $p(x),q(x)in S[X]$ and $p(x)q(x)in S[X]$ for all $p(X)in K[X]$ and all $q(x)in S[X]$ ? Is it the question ?
4) A linear map $f:Gto G'$ is a map s.t. $f(kg)=kf(g)$ for all $kin mathbb Z$ and all $gin G$ and $f(g+h)=f(g)+f(h)$ for all $g,hin G$ ? Is it the question ?
5) $f:Eto E'$ is linear if $f(p(x) y)=p(x)f(y)$ for all $p(x)in K[X]$ and all $yin E$ and $f(y+y')=f(y)+f(y')$ for all $y,y'in E$ ? Is it the question ?
abstract-algebra modules
edited Aug 2 at 2:59
Michael Hardy
204k23185460
asked Aug 1 at 17:37
MathBeginner
695312
edited Aug 2 at 2:59
Michael Hardy
204k23185460
edited Aug 2 at 2:59
Michael Hardy
204k23185460
edited Aug 2 at 2:59
Michael Hardy
204k23185460
204k23185460
asked Aug 1 at 17:37
MathBeginner
695312
asked Aug 1 at 17:37
MathBeginner
695312
asked Aug 1 at 17:37
MathBeginner
695312
695312
3
No. I very much doubt the author of these questions intended you to regurgitate definitions as you are doing. In the first two questions, you can give a simple name to what they have described as submodules. Each one has a familiar and simple alternative name. The third one is not much different: you can describe the submodules because of the special structure of the ideals of $K[X]$ (oops, dropped a hint). I can't help with the 5th because I don't exactly understand what is written.
– rschwieb
Aug 1 at 17:53
@rschwieb If $V$ is a vector space over $K$, and $T in L(V)$, then $V$ together with $T$ can be thought of as a $K[X]$-module where $X$ acts on a vector $v$ by $T$. That is, $Xv=T(v)$. The rest of the action is given from this.
– Jonathan Dunay
Aug 1 at 19:51
add a comment |Â
3
No. I very much doubt the author of these questions intended you to regurgitate definitions as you are doing. In the first two questions, you can give a simple name to what they have described as submodules. Each one has a familiar and simple alternative name. The third one is not much different: you can describe the submodules because of the special structure of the ideals of $K[X]$ (oops, dropped a hint). I can't help with the 5th because I don't exactly understand what is written.
– rschwieb
Aug 1 at 17:53
@rschwieb If $V$ is a vector space over $K$, and $T in L(V)$, then $V$ together with $T$ can be thought of as a $K[X]$-module where $X$ acts on a vector $v$ by $T$. That is, $Xv=T(v)$. The rest of the action is given from this.
– Jonathan Dunay
Aug 1 at 19:51
3
3
No. I very much doubt the author of these questions intended you to regurgitate definitions as you are doing. In the first two questions, you can give a simple name to what they have described as submodules. Each one has a familiar and simple alternative name. The third one is not much different: you can describe the submodules because of the special structure of the ideals of $K[X]$ (oops, dropped a hint). I can't help with the 5th because I don't exactly understand what is written.
– rschwieb
Aug 1 at 17:53
No. I very much doubt the author of these questions intended you to regurgitate definitions as you are doing. In the first two questions, you can give a simple name to what they have described as submodules. Each one has a familiar and simple alternative name. The third one is not much different: you can describe the submodules because of the special structure of the ideals of $K[X]$ (oops, dropped a hint). I can't help with the 5th because I don't exactly understand what is written.
– rschwieb
Aug 1 at 17:53
@rschwieb If $V$ is a vector space over $K$, and $T in L(V)$, then $V$ together with $T$ can be thought of as a $K[X]$-module where $X$ acts on a vector $v$ by $T$. That is, $Xv=T(v)$. The rest of the action is given from this.
– Jonathan Dunay
Aug 1 at 19:51
@rschwieb If $V$ is a vector space over $K$, and $T in L(V)$, then $V$ together with $T$ can be thought of as a $K[X]$-module where $X$ acts on a vector $v$ by $T$. That is, $Xv=T(v)$. The rest of the action is given from this.
– Jonathan Dunay
Aug 1 at 19:51
add a comment |Â
1 Answer
1
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up vote
2
down vote
It appears that the questions are asking you to discuss these in already familiar terms. For example, for 2), the answer would be that the submodules of $G$ considered as a $BbbZ$-module are just the subgroups of $G$.
For the last one, I don't think it is that straightforward. I think they want you to write down a condition on the map $f$ using $u$ and $u'$ that is necessary and sufficient to guarantee that the map is $K[X]$-linear (I know that is vague).
Hint:
If $f$ is a $K$-linear map and for each $y in E$, $f(Xy)=Xf(y)$, then $f$ is $K[X]$-linear.
The condition I have in mind is that:
$f$ is a $K$-linear and $fcirc u = u' circ f$
answered Aug 1 at 19:00
Jonathan Dunay
1,2831215
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
It appears that the questions are asking you to discuss these in already familiar terms. For example, for 2), the answer would be that the submodules of $G$ considered as a $BbbZ$-module are just the subgroups of $G$.
For the last one, I don't think it is that straightforward. I think they want you to write down a condition on the map $f$ using $u$ and $u'$ that is necessary and sufficient to guarantee that the map is $K[X]$-linear (I know that is vague).
Hint:
If $f$ is a $K$-linear map and for each $y in E$, $f(Xy)=Xf(y)$, then $f$ is $K[X]$-linear.
The condition I have in mind is that:
$f$ is a $K$-linear and $fcirc u = u' circ f$
answered Aug 1 at 19:00
Jonathan Dunay
1,2831215
add a comment |Â
up vote
2
down vote
It appears that the questions are asking you to discuss these in already familiar terms. For example, for 2), the answer would be that the submodules of $G$ considered as a $BbbZ$-module are just the subgroups of $G$.
For the last one, I don't think it is that straightforward. I think they want you to write down a condition on the map $f$ using $u$ and $u'$ that is necessary and sufficient to guarantee that the map is $K[X]$-linear (I know that is vague).
Hint:
If $f$ is a $K$-linear map and for each $y in E$, $f(Xy)=Xf(y)$, then $f$ is $K[X]$-linear.
The condition I have in mind is that:
$f$ is a $K$-linear and $fcirc u = u' circ f$
answered Aug 1 at 19:00
Jonathan Dunay
1,2831215
add a comment |Â
up vote
2
down vote
up vote
2
down vote
It appears that the questions are asking you to discuss these in already familiar terms. For example, for 2), the answer would be that the submodules of $G$ considered as a $BbbZ$-module are just the subgroups of $G$.
For the last one, I don't think it is that straightforward. I think they want you to write down a condition on the map $f$ using $u$ and $u'$ that is necessary and sufficient to guarantee that the map is $K[X]$-linear (I know that is vague).
Hint:
If $f$ is a $K$-linear map and for each $y in E$, $f(Xy)=Xf(y)$, then $f$ is $K[X]$-linear.
The condition I have in mind is that:
$f$ is a $K$-linear and $fcirc u = u' circ f$
answered Aug 1 at 19:00
Jonathan Dunay
1,2831215
It appears that the questions are asking you to discuss these in already familiar terms. For example, for 2), the answer would be that the submodules of $G$ considered as a $BbbZ$-module are just the subgroups of $G$.
For the last one, I don't think it is that straightforward. I think they want you to write down a condition on the map $f$ using $u$ and $u'$ that is necessary and sufficient to guarantee that the map is $K[X]$-linear (I know that is vague).
Hint:
If $f$ is a $K$-linear map and for each $y in E$, $f(Xy)=Xf(y)$, then $f$ is $K[X]$-linear.
The condition I have in mind is that:
$f$ is a $K$-linear and $fcirc u = u' circ f$
answered Aug 1 at 19:00
Jonathan Dunay
1,2831215
edited Aug 1 at 19:08
answered Aug 1 at 19:00
Jonathan Dunay
1,2831215
answered Aug 1 at 19:00
Jonathan Dunay
1,2831215
answered Aug 1 at 19:00
Jonathan Dunay
1,2831215
1,2831215
add a comment |Â
add a comment |Â
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No. I very much doubt the author of these questions intended you to regurgitate definitions as you are doing. In the first two questions, you can give a simple name to what they have described as submodules. Each one has a familiar and simple alternative name. The third one is not much different: you can describe the submodules because of the special structure of the ideals of $K[X]$ (oops, dropped a hint). I can't help with the 5th because I don't exactly understand what is written.
– rschwieb
Aug 1 at 17:53
@rschwieb If $V$ is a vector space over $K$, and $T in L(V)$, then $V$ together with $T$ can be thought of as a $K[X]$-module where $X$ acts on a vector $v$ by $T$. That is, $Xv=T(v)$. The rest of the action is given from this.
– Jonathan Dunay
Aug 1 at 19:51