Mixing
Showing that we can't always align the decompositions of finitely generated modules over PID
Clash Royale CLAN TAG#URR8PPP
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I am working on the following exercise.
If $M$ is a finitely generated module over a PID $A$ and $M′$ is a
submodule, can we write $M = F ⊕ T$ and $M′ = F′ ⊕ T′$ such that $F$
and $F′$ are free, $T$ and $T′$ are torsion, and $F′ subset F$ and
$T′ subset T$?
Following a hint, I've assumed $A$ contains an irreducible $p$. I then showed $M′ = A(p, bar1)$ is a free submodule of $M = A ⊕ A/(p)$.
Now I should show that $M'$ cannot be contained in the free part of $M$, but it seems to me that $M'$ is indeed contained in the free part of $M$. I'm not seeing how to use that $p$ is irreducible.
I would appreciate any help.
abstract-algebra modules
asked Jul 21 at 19:44
CuriousKid7
1,438517
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0
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I am working on the following exercise.
If $M$ is a finitely generated module over a PID $A$ and $M′$ is a
submodule, can we write $M = F ⊕ T$ and $M′ = F′ ⊕ T′$ such that $F$
and $F′$ are free, $T$ and $T′$ are torsion, and $F′ subset F$ and
$T′ subset T$?
Following a hint, I've assumed $A$ contains an irreducible $p$. I then showed $M′ = A(p, bar1)$ is a free submodule of $M = A ⊕ A/(p)$.
Now I should show that $M'$ cannot be contained in the free part of $M$, but it seems to me that $M'$ is indeed contained in the free part of $M$. I'm not seeing how to use that $p$ is irreducible.
I would appreciate any help.
abstract-algebra modules
asked Jul 21 at 19:44
CuriousKid7
1,438517
What do you think the free part of $M$ is?
– Eric Wofsey
Jul 21 at 19:47
@EricWofsey The elements $(x, bary)$ where $x$ is nonzero.
– CuriousKid7
Jul 21 at 19:50
Well, perhaps a better question would be, what do you think the definition of "free part" is?
– Eric Wofsey
Jul 21 at 19:51
$ (x, bar y) mid x ne 0 $ isn't a submodule (it doesn't even contain the zero element) so it can't be equal to $F$.
– Daniel Schepler
Jul 21 at 20:42
@DanielSchepler yes true. Is the free part of a module the same as the torsion free elements?
– CuriousKid7
Jul 21 at 21:37
 |Â
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am working on the following exercise.
If $M$ is a finitely generated module over a PID $A$ and $M′$ is a
submodule, can we write $M = F ⊕ T$ and $M′ = F′ ⊕ T′$ such that $F$
and $F′$ are free, $T$ and $T′$ are torsion, and $F′ subset F$ and
$T′ subset T$?
Following a hint, I've assumed $A$ contains an irreducible $p$. I then showed $M′ = A(p, bar1)$ is a free submodule of $M = A ⊕ A/(p)$.
Now I should show that $M'$ cannot be contained in the free part of $M$, but it seems to me that $M'$ is indeed contained in the free part of $M$. I'm not seeing how to use that $p$ is irreducible.
I would appreciate any help.
abstract-algebra modules
asked Jul 21 at 19:44
CuriousKid7
1,438517
I am working on the following exercise.
If $M$ is a finitely generated module over a PID $A$ and $M′$ is a
submodule, can we write $M = F ⊕ T$ and $M′ = F′ ⊕ T′$ such that $F$
and $F′$ are free, $T$ and $T′$ are torsion, and $F′ subset F$ and
$T′ subset T$?
Following a hint, I've assumed $A$ contains an irreducible $p$. I then showed $M′ = A(p, bar1)$ is a free submodule of $M = A ⊕ A/(p)$.
Now I should show that $M'$ cannot be contained in the free part of $M$, but it seems to me that $M'$ is indeed contained in the free part of $M$. I'm not seeing how to use that $p$ is irreducible.
I would appreciate any help.
abstract-algebra modules
asked Jul 21 at 19:44
CuriousKid7
1,438517
asked Jul 21 at 19:44
CuriousKid7
1,438517
asked Jul 21 at 19:44
CuriousKid7
1,438517
asked Jul 21 at 19:44
CuriousKid7
1,438517
1,438517
What do you think the free part of $M$ is?
– Eric Wofsey
Jul 21 at 19:47
@EricWofsey The elements $(x, bary)$ where $x$ is nonzero.
– CuriousKid7
Jul 21 at 19:50
Well, perhaps a better question would be, what do you think the definition of "free part" is?
– Eric Wofsey
Jul 21 at 19:51
$ (x, bar y) mid x ne 0 $ isn't a submodule (it doesn't even contain the zero element) so it can't be equal to $F$.
– Daniel Schepler
Jul 21 at 20:42
@DanielSchepler yes true. Is the free part of a module the same as the torsion free elements?
– CuriousKid7
Jul 21 at 21:37
 |Â
show 1 more comment
What do you think the free part of $M$ is?
– Eric Wofsey
Jul 21 at 19:47
@EricWofsey The elements $(x, bary)$ where $x$ is nonzero.
– CuriousKid7
Jul 21 at 19:50
Well, perhaps a better question would be, what do you think the definition of "free part" is?
– Eric Wofsey
Jul 21 at 19:51
$ (x, bar y) mid x ne 0 $ isn't a submodule (it doesn't even contain the zero element) so it can't be equal to $F$.
– Daniel Schepler
Jul 21 at 20:42
@DanielSchepler yes true. Is the free part of a module the same as the torsion free elements?
– CuriousKid7
Jul 21 at 21:37
What do you think the free part of $M$ is?
– Eric Wofsey
Jul 21 at 19:47
What do you think the free part of $M$ is?
– Eric Wofsey
Jul 21 at 19:47
@EricWofsey The elements $(x, bary)$ where $x$ is nonzero.
– CuriousKid7
Jul 21 at 19:50
@EricWofsey The elements $(x, bary)$ where $x$ is nonzero.
– CuriousKid7
Jul 21 at 19:50
Well, perhaps a better question would be, what do you think the definition of "free part" is?
– Eric Wofsey
Jul 21 at 19:51
Well, perhaps a better question would be, what do you think the definition of "free part" is?
– Eric Wofsey
Jul 21 at 19:51
$ (x, bar y) mid x ne 0 $ isn't a submodule (it doesn't even contain the zero element) so it can't be equal to $F$.
– Daniel Schepler
Jul 21 at 20:42
$ (x, bar y) mid x ne 0 $ isn't a submodule (it doesn't even contain the zero element) so it can't be equal to $F$.
– Daniel Schepler
Jul 21 at 20:42
@DanielSchepler yes true. Is the free part of a module the same as the torsion free elements?
– CuriousKid7
Jul 21 at 21:37
@DanielSchepler yes true. Is the free part of a module the same as the torsion free elements?
– CuriousKid7
Jul 21 at 21:37
 |Â
show 1 more comment
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What do you think the free part of $M$ is?
– Eric Wofsey
Jul 21 at 19:47
@EricWofsey The elements $(x, bary)$ where $x$ is nonzero.
– CuriousKid7
Jul 21 at 19:50
Well, perhaps a better question would be, what do you think the definition of "free part" is?
– Eric Wofsey
Jul 21 at 19:51
$ (x, bar y) mid x ne 0 $ isn't a submodule (it doesn't even contain the zero element) so it can't be equal to $F$.
– Daniel Schepler
Jul 21 at 20:42
@DanielSchepler yes true. Is the free part of a module the same as the torsion free elements?
– CuriousKid7
Jul 21 at 21:37