References on the submodules of a direct sum of copies of $mathbbZ$Why is cofiniteness included in the definition of direct sum of submodules?Two submodules $H$ and $H^*$ that are both direct factors but $H cap H^*$ is not a direct factor Group $mathbb Q^*$ as direct product/sumCan any torsion free abelian group be embedded in a direct sum of copies of $mathbb Q$?The rationals as a direct summand of the realsInfinitely many direct sum decompositions of $M$ into direct sum of irreducible $mathbbCG$-modules?All $mathbbZ$-submodules of $mathbbZoplusmathbbZ$ are free.Decomposition of finite KG-Module in direct Sum of Submodules for a cyclic Group GEndomorphisms of $mathbb Z$-modules are the same as those of $mathbb F_p$-vector spacesFind the pure submodules of a cyclic $mathbbZ$-module of order 12
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References on the submodules of a direct sum of copies of $mathbbZ$
Why is cofiniteness included in the definition of direct sum of submodules?Two submodules $H$ and $H^*$ that are both direct factors but $H cap H^*$ is not a direct factor Group $mathbb Q^*$ as direct product/sumCan any torsion free abelian group be embedded in a direct sum of copies of $mathbb Q$?The rationals as a direct summand of the realsInfinitely many direct sum decompositions of $M$ into direct sum of irreducible $mathbbCG$-modules?All $mathbbZ$-submodules of $mathbbZoplusmathbbZ$ are free.Decomposition of finite KG-Module in direct Sum of Submodules for a cyclic Group GEndomorphisms of $mathbb Z$-modules are the same as those of $mathbb F_p$-vector spacesFind the pure submodules of a cyclic $mathbbZ$-module of order 12
$begingroup$
Let $mathbbP$ be the set of all prime numbers. Consider the $mathbbZ$-Module $mathbbZ^(mathbbP)$, that is, the external direct sum of copies of the additive abelian group $mathbbZ$ indexed in the set of prime numbers.
Does anybody know any books or articles where I can read about this object? In particular, I'm looking for any characterization of it's submodules.
Thanks in advance for any help provided.
abstract-algebra reference-request modules direct-sum
asked Mar 11 at 21:08
eNReNR
13310
$endgroup$
add a comment |
$begingroup$
Let $mathbbP$ be the set of all prime numbers. Consider the $mathbbZ$-Module $mathbbZ^(mathbbP)$, that is, the external direct sum of copies of the additive abelian group $mathbbZ$ indexed in the set of prime numbers.
Does anybody know any books or articles where I can read about this object? In particular, I'm looking for any characterization of it's submodules.
Thanks in advance for any help provided.
abstract-algebra reference-request modules direct-sum
asked Mar 11 at 21:08
eNReNR
13310
$endgroup$
1
$begingroup$
It is canonically isomorphic to $BbbQ^times$. Submodules are simply subgroups.
$endgroup$
– Servaes
Mar 11 at 22:15
1
$begingroup$
@Servaes: almost; you need to quotient out by $-1$.
$endgroup$
– Qiaochu Yuan
Mar 12 at 1:48
$begingroup$
@QiaochuYuan Sharp as always, too late to edit unfortunately.
$endgroup$
– Servaes
Mar 12 at 1:58
add a comment |
$begingroup$
Let $mathbbP$ be the set of all prime numbers. Consider the $mathbbZ$-Module $mathbbZ^(mathbbP)$, that is, the external direct sum of copies of the additive abelian group $mathbbZ$ indexed in the set of prime numbers.
Does anybody know any books or articles where I can read about this object? In particular, I'm looking for any characterization of it's submodules.
Thanks in advance for any help provided.
abstract-algebra reference-request modules direct-sum
asked Mar 11 at 21:08
eNReNR
13310
$endgroup$
Let $mathbbP$ be the set of all prime numbers. Consider the $mathbbZ$-Module $mathbbZ^(mathbbP)$, that is, the external direct sum of copies of the additive abelian group $mathbbZ$ indexed in the set of prime numbers.
Does anybody know any books or articles where I can read about this object? In particular, I'm looking for any characterization of it's submodules.
Thanks in advance for any help provided.
abstract-algebra reference-request modules direct-sum
abstract-algebra reference-request modules direct-sum
asked Mar 11 at 21:08
eNReNR
13310
asked Mar 11 at 21:08
eNReNR
13310
asked Mar 11 at 21:08
eNReNR
13310
asked Mar 11 at 21:08
eNReNR
13310
asked Mar 11 at 21:08
eNReNR
13310
13310
1
$begingroup$
It is canonically isomorphic to $BbbQ^times$. Submodules are simply subgroups.
$endgroup$
– Servaes
Mar 11 at 22:15
1
$begingroup$
@Servaes: almost; you need to quotient out by $-1$.
$endgroup$
– Qiaochu Yuan
Mar 12 at 1:48
$begingroup$
@QiaochuYuan Sharp as always, too late to edit unfortunately.
$endgroup$
– Servaes
Mar 12 at 1:58
add a comment |
1
$begingroup$
It is canonically isomorphic to $BbbQ^times$. Submodules are simply subgroups.
$endgroup$
– Servaes
Mar 11 at 22:15
1
$begingroup$
@Servaes: almost; you need to quotient out by $-1$.
$endgroup$
– Qiaochu Yuan
Mar 12 at 1:48
$begingroup$
@QiaochuYuan Sharp as always, too late to edit unfortunately.
$endgroup$
– Servaes
Mar 12 at 1:58
1
1
$begingroup$
It is canonically isomorphic to $BbbQ^times$. Submodules are simply subgroups.
$endgroup$
– Servaes
Mar 11 at 22:15
$begingroup$
It is canonically isomorphic to $BbbQ^times$. Submodules are simply subgroups.
$endgroup$
– Servaes
Mar 11 at 22:15
1
1
$begingroup$
@Servaes: almost; you need to quotient out by $-1$.
$endgroup$
– Qiaochu Yuan
Mar 12 at 1:48
$begingroup$
@Servaes: almost; you need to quotient out by $-1$.
$endgroup$
– Qiaochu Yuan
Mar 12 at 1:48
$begingroup$
@QiaochuYuan Sharp as always, too late to edit unfortunately.
$endgroup$
– Servaes
Mar 12 at 1:58
$begingroup$
@QiaochuYuan Sharp as always, too late to edit unfortunately.
$endgroup$
– Servaes
Mar 12 at 1:58
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
As in Servaes' comment, presumably this question was inspired by trying to understand the multiplicative group $mathbbQ^times$ of the rationals. Note that abstractly you don't need to know anything about $mathbbP$ other than that it's countably infinite, so it's equivalent to just study the direct sum $bigoplus_i in mathbbN mathbbZ$ of countably many copies of $mathbbZ$.
This is a free module, and submodules of free modules over a PID are free. So every submodule is the span of some linearly independent set of elements. I'm not sure what else there is to say other than this.
answered Mar 12 at 1:51
Qiaochu YuanQiaochu Yuan
281k32592938
$endgroup$
$begingroup$
First of all, thank you for your answer. That being said, I would argue that there is something in the nature of this direct sum that makes it different from $bigoplus_ninmathbbNmathbbZ$ since in this article (sciencedirect.com/science/article/pii/S0021869316000612) they use it for an example, and the authors specifically use the sum I asked about. Naturally I thoguht about using $mathbbN$ as the indexing set, but unfortunately I was unable to prove that either were an example of the objects that are being treated in the article.
$endgroup$
– eNR
Mar 12 at 3:28
$begingroup$
There is no difference; any bijection between $mathbbN$ and the prime numbers exhibits an isomorphism between them. Can you quote the relevant section of that article? I don't have access to it.
$endgroup$
– Qiaochu Yuan
Mar 12 at 8:36
$begingroup$
They state that $mathbbZ^(mathbbP)bigoplusmathbbZ$ is an example of an isoartinian module. We say that an $R$-Module $M$ is isoartinian iff for every nonempty set $Gamma$ of submodules of $M$, there exists $NinGamma$ such that, for every $N'leq N$, if $N'inGamma$, then $N'cong N$.
$endgroup$
– eNR
Mar 13 at 2:42
add a comment |
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1 Answer
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1 Answer
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votes
$begingroup$
As in Servaes' comment, presumably this question was inspired by trying to understand the multiplicative group $mathbbQ^times$ of the rationals. Note that abstractly you don't need to know anything about $mathbbP$ other than that it's countably infinite, so it's equivalent to just study the direct sum $bigoplus_i in mathbbN mathbbZ$ of countably many copies of $mathbbZ$.
This is a free module, and submodules of free modules over a PID are free. So every submodule is the span of some linearly independent set of elements. I'm not sure what else there is to say other than this.
answered Mar 12 at 1:51
Qiaochu YuanQiaochu Yuan
281k32592938
$endgroup$
$begingroup$
First of all, thank you for your answer. That being said, I would argue that there is something in the nature of this direct sum that makes it different from $bigoplus_ninmathbbNmathbbZ$ since in this article (sciencedirect.com/science/article/pii/S0021869316000612) they use it for an example, and the authors specifically use the sum I asked about. Naturally I thoguht about using $mathbbN$ as the indexing set, but unfortunately I was unable to prove that either were an example of the objects that are being treated in the article.
$endgroup$
– eNR
Mar 12 at 3:28
$begingroup$
There is no difference; any bijection between $mathbbN$ and the prime numbers exhibits an isomorphism between them. Can you quote the relevant section of that article? I don't have access to it.
$endgroup$
– Qiaochu Yuan
Mar 12 at 8:36
$begingroup$
They state that $mathbbZ^(mathbbP)bigoplusmathbbZ$ is an example of an isoartinian module. We say that an $R$-Module $M$ is isoartinian iff for every nonempty set $Gamma$ of submodules of $M$, there exists $NinGamma$ such that, for every $N'leq N$, if $N'inGamma$, then $N'cong N$.
$endgroup$
– eNR
Mar 13 at 2:42
add a comment |
$begingroup$
As in Servaes' comment, presumably this question was inspired by trying to understand the multiplicative group $mathbbQ^times$ of the rationals. Note that abstractly you don't need to know anything about $mathbbP$ other than that it's countably infinite, so it's equivalent to just study the direct sum $bigoplus_i in mathbbN mathbbZ$ of countably many copies of $mathbbZ$.
This is a free module, and submodules of free modules over a PID are free. So every submodule is the span of some linearly independent set of elements. I'm not sure what else there is to say other than this.
answered Mar 12 at 1:51
Qiaochu YuanQiaochu Yuan
281k32592938
$endgroup$
$begingroup$
First of all, thank you for your answer. That being said, I would argue that there is something in the nature of this direct sum that makes it different from $bigoplus_ninmathbbNmathbbZ$ since in this article (sciencedirect.com/science/article/pii/S0021869316000612) they use it for an example, and the authors specifically use the sum I asked about. Naturally I thoguht about using $mathbbN$ as the indexing set, but unfortunately I was unable to prove that either were an example of the objects that are being treated in the article.
$endgroup$
– eNR
Mar 12 at 3:28
$begingroup$
There is no difference; any bijection between $mathbbN$ and the prime numbers exhibits an isomorphism between them. Can you quote the relevant section of that article? I don't have access to it.
$endgroup$
– Qiaochu Yuan
Mar 12 at 8:36
$begingroup$
They state that $mathbbZ^(mathbbP)bigoplusmathbbZ$ is an example of an isoartinian module. We say that an $R$-Module $M$ is isoartinian iff for every nonempty set $Gamma$ of submodules of $M$, there exists $NinGamma$ such that, for every $N'leq N$, if $N'inGamma$, then $N'cong N$.
$endgroup$
– eNR
Mar 13 at 2:42
add a comment |
$begingroup$
As in Servaes' comment, presumably this question was inspired by trying to understand the multiplicative group $mathbbQ^times$ of the rationals. Note that abstractly you don't need to know anything about $mathbbP$ other than that it's countably infinite, so it's equivalent to just study the direct sum $bigoplus_i in mathbbN mathbbZ$ of countably many copies of $mathbbZ$.
This is a free module, and submodules of free modules over a PID are free. So every submodule is the span of some linearly independent set of elements. I'm not sure what else there is to say other than this.
answered Mar 12 at 1:51
Qiaochu YuanQiaochu Yuan
281k32592938
$endgroup$
As in Servaes' comment, presumably this question was inspired by trying to understand the multiplicative group $mathbbQ^times$ of the rationals. Note that abstractly you don't need to know anything about $mathbbP$ other than that it's countably infinite, so it's equivalent to just study the direct sum $bigoplus_i in mathbbN mathbbZ$ of countably many copies of $mathbbZ$.
This is a free module, and submodules of free modules over a PID are free. So every submodule is the span of some linearly independent set of elements. I'm not sure what else there is to say other than this.
answered Mar 12 at 1:51
Qiaochu YuanQiaochu Yuan
281k32592938
answered Mar 12 at 1:51
Qiaochu YuanQiaochu Yuan
281k32592938
answered Mar 12 at 1:51
Qiaochu YuanQiaochu Yuan
281k32592938
answered Mar 12 at 1:51
Qiaochu YuanQiaochu Yuan
281k32592938
281k32592938
$begingroup$
First of all, thank you for your answer. That being said, I would argue that there is something in the nature of this direct sum that makes it different from $bigoplus_ninmathbbNmathbbZ$ since in this article (sciencedirect.com/science/article/pii/S0021869316000612) they use it for an example, and the authors specifically use the sum I asked about. Naturally I thoguht about using $mathbbN$ as the indexing set, but unfortunately I was unable to prove that either were an example of the objects that are being treated in the article.
$endgroup$
– eNR
Mar 12 at 3:28
$begingroup$
There is no difference; any bijection between $mathbbN$ and the prime numbers exhibits an isomorphism between them. Can you quote the relevant section of that article? I don't have access to it.
$endgroup$
– Qiaochu Yuan
Mar 12 at 8:36
$begingroup$
They state that $mathbbZ^(mathbbP)bigoplusmathbbZ$ is an example of an isoartinian module. We say that an $R$-Module $M$ is isoartinian iff for every nonempty set $Gamma$ of submodules of $M$, there exists $NinGamma$ such that, for every $N'leq N$, if $N'inGamma$, then $N'cong N$.
$endgroup$
– eNR
Mar 13 at 2:42
add a comment |
$begingroup$
First of all, thank you for your answer. That being said, I would argue that there is something in the nature of this direct sum that makes it different from $bigoplus_ninmathbbNmathbbZ$ since in this article (sciencedirect.com/science/article/pii/S0021869316000612) they use it for an example, and the authors specifically use the sum I asked about. Naturally I thoguht about using $mathbbN$ as the indexing set, but unfortunately I was unable to prove that either were an example of the objects that are being treated in the article.
$endgroup$
– eNR
Mar 12 at 3:28
$begingroup$
There is no difference; any bijection between $mathbbN$ and the prime numbers exhibits an isomorphism between them. Can you quote the relevant section of that article? I don't have access to it.
$endgroup$
– Qiaochu Yuan
Mar 12 at 8:36
$begingroup$
They state that $mathbbZ^(mathbbP)bigoplusmathbbZ$ is an example of an isoartinian module. We say that an $R$-Module $M$ is isoartinian iff for every nonempty set $Gamma$ of submodules of $M$, there exists $NinGamma$ such that, for every $N'leq N$, if $N'inGamma$, then $N'cong N$.
$endgroup$
– eNR
Mar 13 at 2:42
$begingroup$
First of all, thank you for your answer. That being said, I would argue that there is something in the nature of this direct sum that makes it different from $bigoplus_ninmathbbNmathbbZ$ since in this article (sciencedirect.com/science/article/pii/S0021869316000612) they use it for an example, and the authors specifically use the sum I asked about. Naturally I thoguht about using $mathbbN$ as the indexing set, but unfortunately I was unable to prove that either were an example of the objects that are being treated in the article.
$endgroup$
– eNR
Mar 12 at 3:28
$begingroup$
First of all, thank you for your answer. That being said, I would argue that there is something in the nature of this direct sum that makes it different from $bigoplus_ninmathbbNmathbbZ$ since in this article (sciencedirect.com/science/article/pii/S0021869316000612) they use it for an example, and the authors specifically use the sum I asked about. Naturally I thoguht about using $mathbbN$ as the indexing set, but unfortunately I was unable to prove that either were an example of the objects that are being treated in the article.
$endgroup$
– eNR
Mar 12 at 3:28
$begingroup$
There is no difference; any bijection between $mathbbN$ and the prime numbers exhibits an isomorphism between them. Can you quote the relevant section of that article? I don't have access to it.
$endgroup$
– Qiaochu Yuan
Mar 12 at 8:36
$begingroup$
There is no difference; any bijection between $mathbbN$ and the prime numbers exhibits an isomorphism between them. Can you quote the relevant section of that article? I don't have access to it.
$endgroup$
– Qiaochu Yuan
Mar 12 at 8:36
$begingroup$
They state that $mathbbZ^(mathbbP)bigoplusmathbbZ$ is an example of an isoartinian module. We say that an $R$-Module $M$ is isoartinian iff for every nonempty set $Gamma$ of submodules of $M$, there exists $NinGamma$ such that, for every $N'leq N$, if $N'inGamma$, then $N'cong N$.
$endgroup$
– eNR
Mar 13 at 2:42
$begingroup$
They state that $mathbbZ^(mathbbP)bigoplusmathbbZ$ is an example of an isoartinian module. We say that an $R$-Module $M$ is isoartinian iff for every nonempty set $Gamma$ of submodules of $M$, there exists $NinGamma$ such that, for every $N'leq N$, if $N'inGamma$, then $N'cong N$.
$endgroup$
– eNR
Mar 13 at 2:42
add a comment |
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$begingroup$
It is canonically isomorphic to $BbbQ^times$. Submodules are simply subgroups.
$endgroup$
– Servaes
Mar 11 at 22:15
1
$begingroup$
@Servaes: almost; you need to quotient out by $-1$.
$endgroup$
– Qiaochu Yuan
Mar 12 at 1:48
$begingroup$
@QiaochuYuan Sharp as always, too late to edit unfortunately.
$endgroup$
– Servaes
Mar 12 at 1:58