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Please clarify the meaning of $textEnd_R(R_R, R_R) cong R$ [closed]
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First of all, I have no idea what $textEnd_R(R_R, R_R) cong R$ is even supposed to mean. The $R$ in $textEnd_R$ is supposed to be a category, but the $R$ in $R_R$ is an object in that category. So I don't even know what the question is asking.
ring-theory notation modules
asked Jul 17 at 0:17
Tomislav Ostojich
475313
closed as unclear what you're asking by Alex Provost, amWhy, Adrian Keister, Trần Thúc Minh TrÃ, Parcly Taxel Jul 17 at 2:43
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |Â
up vote
-2
down vote
favorite
First of all, I have no idea what $textEnd_R(R_R, R_R) cong R$ is even supposed to mean. The $R$ in $textEnd_R$ is supposed to be a category, but the $R$ in $R_R$ is an object in that category. So I don't even know what the question is asking.
ring-theory notation modules
asked Jul 17 at 0:17
Tomislav Ostojich
475313
closed as unclear what you're asking by Alex Provost, amWhy, Adrian Keister, Trần Thúc Minh TrÃ, Parcly Taxel Jul 17 at 2:43
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
2
Your title indicates that you are asking for a proof, but your question indicates that you are asking for a clarification of notation. Which is it?
– Y. Forman
Jul 17 at 0:22
I am asking for a clarification of notation.
– Tomislav Ostojich
Jul 17 at 0:23
You should give a citation or other clarification to where you see this notation being used. Almost always an author will introduce and define notation before using it, by very nature of a useful notation being a compact way to avoid repeating some basic ideas over and over (the ability of skillful notation to hide details while suggesting intuitive properties).
– hardmath
Jul 17 at 0:31
It seems likely the user (or whoever is authoring the user's material) is confusing $Hom(R_R,R_R)=End(R_R)$. Personally I like the convention of specifying the rings of interest in the End notation as subscripts. So for example if $_RM_S$ and $_RN_S$ were $R,S$ bimodules, and I wanted to write the set of $S$ linear maps between them, I would say $Hom(M_S, N_S)$. As far as I can tell this eliminates the need for another subscript outside the parens.
– rschwieb
Jul 19 at 19:09
add a comment |Â
up vote
-2
down vote
favorite
up vote
-2
down vote
favorite
First of all, I have no idea what $textEnd_R(R_R, R_R) cong R$ is even supposed to mean. The $R$ in $textEnd_R$ is supposed to be a category, but the $R$ in $R_R$ is an object in that category. So I don't even know what the question is asking.
ring-theory notation modules
asked Jul 17 at 0:17
Tomislav Ostojich
475313
First of all, I have no idea what $textEnd_R(R_R, R_R) cong R$ is even supposed to mean. The $R$ in $textEnd_R$ is supposed to be a category, but the $R$ in $R_R$ is an object in that category. So I don't even know what the question is asking.
ring-theory notation modules
asked Jul 17 at 0:17
Tomislav Ostojich
475313
edited Jul 17 at 0:31
asked Jul 17 at 0:17
Tomislav Ostojich
475313
asked Jul 17 at 0:17
Tomislav Ostojich
475313
asked Jul 17 at 0:17
Tomislav Ostojich
475313
475313
closed as unclear what you're asking by Alex Provost, amWhy, Adrian Keister, Trần Thúc Minh TrÃ, Parcly Taxel Jul 17 at 2:43
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Alex Provost, amWhy, Adrian Keister, Trần Thúc Minh TrÃ, Parcly Taxel Jul 17 at 2:43
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
2
Your title indicates that you are asking for a proof, but your question indicates that you are asking for a clarification of notation. Which is it?
– Y. Forman
Jul 17 at 0:22
I am asking for a clarification of notation.
– Tomislav Ostojich
Jul 17 at 0:23
You should give a citation or other clarification to where you see this notation being used. Almost always an author will introduce and define notation before using it, by very nature of a useful notation being a compact way to avoid repeating some basic ideas over and over (the ability of skillful notation to hide details while suggesting intuitive properties).
– hardmath
Jul 17 at 0:31
It seems likely the user (or whoever is authoring the user's material) is confusing $Hom(R_R,R_R)=End(R_R)$. Personally I like the convention of specifying the rings of interest in the End notation as subscripts. So for example if $_RM_S$ and $_RN_S$ were $R,S$ bimodules, and I wanted to write the set of $S$ linear maps between them, I would say $Hom(M_S, N_S)$. As far as I can tell this eliminates the need for another subscript outside the parens.
– rschwieb
Jul 19 at 19:09
add a comment |Â
2
Your title indicates that you are asking for a proof, but your question indicates that you are asking for a clarification of notation. Which is it?
– Y. Forman
Jul 17 at 0:22
I am asking for a clarification of notation.
– Tomislav Ostojich
Jul 17 at 0:23
You should give a citation or other clarification to where you see this notation being used. Almost always an author will introduce and define notation before using it, by very nature of a useful notation being a compact way to avoid repeating some basic ideas over and over (the ability of skillful notation to hide details while suggesting intuitive properties).
– hardmath
Jul 17 at 0:31
It seems likely the user (or whoever is authoring the user's material) is confusing $Hom(R_R,R_R)=End(R_R)$. Personally I like the convention of specifying the rings of interest in the End notation as subscripts. So for example if $_RM_S$ and $_RN_S$ were $R,S$ bimodules, and I wanted to write the set of $S$ linear maps between them, I would say $Hom(M_S, N_S)$. As far as I can tell this eliminates the need for another subscript outside the parens.
– rschwieb
Jul 19 at 19:09
2
2
Your title indicates that you are asking for a proof, but your question indicates that you are asking for a clarification of notation. Which is it?
– Y. Forman
Jul 17 at 0:22
Your title indicates that you are asking for a proof, but your question indicates that you are asking for a clarification of notation. Which is it?
– Y. Forman
Jul 17 at 0:22
I am asking for a clarification of notation.
– Tomislav Ostojich
Jul 17 at 0:23
I am asking for a clarification of notation.
– Tomislav Ostojich
Jul 17 at 0:23
You should give a citation or other clarification to where you see this notation being used. Almost always an author will introduce and define notation before using it, by very nature of a useful notation being a compact way to avoid repeating some basic ideas over and over (the ability of skillful notation to hide details while suggesting intuitive properties).
– hardmath
Jul 17 at 0:31
You should give a citation or other clarification to where you see this notation being used. Almost always an author will introduce and define notation before using it, by very nature of a useful notation being a compact way to avoid repeating some basic ideas over and over (the ability of skillful notation to hide details while suggesting intuitive properties).
– hardmath
Jul 17 at 0:31
It seems likely the user (or whoever is authoring the user's material) is confusing $Hom(R_R,R_R)=End(R_R)$. Personally I like the convention of specifying the rings of interest in the End notation as subscripts. So for example if $_RM_S$ and $_RN_S$ were $R,S$ bimodules, and I wanted to write the set of $S$ linear maps between them, I would say $Hom(M_S, N_S)$. As far as I can tell this eliminates the need for another subscript outside the parens.
– rschwieb
Jul 19 at 19:09
It seems likely the user (or whoever is authoring the user's material) is confusing $Hom(R_R,R_R)=End(R_R)$. Personally I like the convention of specifying the rings of interest in the End notation as subscripts. So for example if $_RM_S$ and $_RN_S$ were $R,S$ bimodules, and I wanted to write the set of $S$ linear maps between them, I would say $Hom(M_S, N_S)$. As far as I can tell this eliminates the need for another subscript outside the parens.
– rschwieb
Jul 19 at 19:09
add a comment |Â
1 Answer
1
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1
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accepted
I agree with hardmath's comment that notation should be referenced in context, but here's my best guess.
$R$ is a ring. $R_R$ is a notation for the ring $R$, considered as a module over itself (i.e., as an object in the category $R-textbfmod$). $mathrmEnd_R$ refers the endomorphism ring of endomorphisms in the category $R-textbfmod$; i.e., the $R$ subscript in $mathrmEnd_R$ is short for $R-textbfmod$. As the endomorphism ring and $R$ are both rings, the isomorphism in question in as isomorphism of rings.
answered Jul 17 at 0:50
Y. Forman
10.8k323
It might help to clarify the claimed isomorphism to specify that $R$ is a ring with unity, which simplifies the characterization of endomorphisms $Rto R$ (as $R$-modules).
– hardmath
Jul 17 at 1:20
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
I agree with hardmath's comment that notation should be referenced in context, but here's my best guess.
$R$ is a ring. $R_R$ is a notation for the ring $R$, considered as a module over itself (i.e., as an object in the category $R-textbfmod$). $mathrmEnd_R$ refers the endomorphism ring of endomorphisms in the category $R-textbfmod$; i.e., the $R$ subscript in $mathrmEnd_R$ is short for $R-textbfmod$. As the endomorphism ring and $R$ are both rings, the isomorphism in question in as isomorphism of rings.
answered Jul 17 at 0:50
Y. Forman
10.8k323
It might help to clarify the claimed isomorphism to specify that $R$ is a ring with unity, which simplifies the characterization of endomorphisms $Rto R$ (as $R$-modules).
– hardmath
Jul 17 at 1:20
add a comment |Â
up vote
1
down vote
accepted
I agree with hardmath's comment that notation should be referenced in context, but here's my best guess.
$R$ is a ring. $R_R$ is a notation for the ring $R$, considered as a module over itself (i.e., as an object in the category $R-textbfmod$). $mathrmEnd_R$ refers the endomorphism ring of endomorphisms in the category $R-textbfmod$; i.e., the $R$ subscript in $mathrmEnd_R$ is short for $R-textbfmod$. As the endomorphism ring and $R$ are both rings, the isomorphism in question in as isomorphism of rings.
answered Jul 17 at 0:50
Y. Forman
10.8k323
It might help to clarify the claimed isomorphism to specify that $R$ is a ring with unity, which simplifies the characterization of endomorphisms $Rto R$ (as $R$-modules).
– hardmath
Jul 17 at 1:20
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
I agree with hardmath's comment that notation should be referenced in context, but here's my best guess.
$R$ is a ring. $R_R$ is a notation for the ring $R$, considered as a module over itself (i.e., as an object in the category $R-textbfmod$). $mathrmEnd_R$ refers the endomorphism ring of endomorphisms in the category $R-textbfmod$; i.e., the $R$ subscript in $mathrmEnd_R$ is short for $R-textbfmod$. As the endomorphism ring and $R$ are both rings, the isomorphism in question in as isomorphism of rings.
answered Jul 17 at 0:50
Y. Forman
10.8k323
I agree with hardmath's comment that notation should be referenced in context, but here's my best guess.
$R$ is a ring. $R_R$ is a notation for the ring $R$, considered as a module over itself (i.e., as an object in the category $R-textbfmod$). $mathrmEnd_R$ refers the endomorphism ring of endomorphisms in the category $R-textbfmod$; i.e., the $R$ subscript in $mathrmEnd_R$ is short for $R-textbfmod$. As the endomorphism ring and $R$ are both rings, the isomorphism in question in as isomorphism of rings.
answered Jul 17 at 0:50
Y. Forman
10.8k323
answered Jul 17 at 0:50
Y. Forman
10.8k323
answered Jul 17 at 0:50
Y. Forman
10.8k323
answered Jul 17 at 0:50
Y. Forman
10.8k323
10.8k323
It might help to clarify the claimed isomorphism to specify that $R$ is a ring with unity, which simplifies the characterization of endomorphisms $Rto R$ (as $R$-modules).
– hardmath
Jul 17 at 1:20
add a comment |Â
It might help to clarify the claimed isomorphism to specify that $R$ is a ring with unity, which simplifies the characterization of endomorphisms $Rto R$ (as $R$-modules).
– hardmath
Jul 17 at 1:20
It might help to clarify the claimed isomorphism to specify that $R$ is a ring with unity, which simplifies the characterization of endomorphisms $Rto R$ (as $R$-modules).
– hardmath
Jul 17 at 1:20
It might help to clarify the claimed isomorphism to specify that $R$ is a ring with unity, which simplifies the characterization of endomorphisms $Rto R$ (as $R$-modules).
– hardmath
Jul 17 at 1:20
add a comment |Â
2
Your title indicates that you are asking for a proof, but your question indicates that you are asking for a clarification of notation. Which is it?
– Y. Forman
Jul 17 at 0:22
I am asking for a clarification of notation.
– Tomislav Ostojich
Jul 17 at 0:23
You should give a citation or other clarification to where you see this notation being used. Almost always an author will introduce and define notation before using it, by very nature of a useful notation being a compact way to avoid repeating some basic ideas over and over (the ability of skillful notation to hide details while suggesting intuitive properties).
– hardmath
Jul 17 at 0:31
It seems likely the user (or whoever is authoring the user's material) is confusing $Hom(R_R,R_R)=End(R_R)$. Personally I like the convention of specifying the rings of interest in the End notation as subscripts. So for example if $_RM_S$ and $_RN_S$ were $R,S$ bimodules, and I wanted to write the set of $S$ linear maps between them, I would say $Hom(M_S, N_S)$. As far as I can tell this eliminates the need for another subscript outside the parens.
– rschwieb
Jul 19 at 19:09