Linear dependence as a binary relation
$begingroup$
Elements $a$ and $b$ from an $R$-module $M$ are linearly dependent if there are scalars $x$, $y$ in $R$, $x neq 0$ or $y neq 0$, such that $xa = yb$.
Let $M$ be a torsion-free module over an integral domain $R$.
In this case the binary relation has the following properties:
1. It is an equivalence relation on nonzero elements of $M$;
2. Each equivalence class combined with the zero element is a submodule of $M$.
Let's call the submodule an axis of module $M$.
Clearly, if $M$ is a free module over a principal ideal domain, all the axes are lines isomorphic to $R$ over itself.
But what if $M$ is not a free module or $R$ is not a PID?
There is an example of a torsion-free module over an integral domain where the only axis is the module itself, and it is not a line: $langle 2,x rangle$ over $mathbb{Z}[x]$ (Show that $langle 2,x rangle$ is not a principal ideal in $mathbb Z [x]$).
Questions:
1. Are the axes of a torsion-free module over an integral domain isomorphic to each other?
2. Are the axes of a free module over an integral domain lines?
3. If an axis $A$ of a free module over an integral domain is a line, is there a direct sum decomposition $M = A oplus M'$?
abstract-algebra modules direct-sum free-modules
asked Jan 9 at 6:56
Alex CAlex C
10718
$endgroup$
add a comment |
$begingroup$
Elements $a$ and $b$ from an $R$-module $M$ are linearly dependent if there are scalars $x$, $y$ in $R$, $x neq 0$ or $y neq 0$, such that $xa = yb$.
Let $M$ be a torsion-free module over an integral domain $R$.
In this case the binary relation has the following properties:
1. It is an equivalence relation on nonzero elements of $M$;
2. Each equivalence class combined with the zero element is a submodule of $M$.
Let's call the submodule an axis of module $M$.
Clearly, if $M$ is a free module over a principal ideal domain, all the axes are lines isomorphic to $R$ over itself.
But what if $M$ is not a free module or $R$ is not a PID?
There is an example of a torsion-free module over an integral domain where the only axis is the module itself, and it is not a line: $langle 2,x rangle$ over $mathbb{Z}[x]$ (Show that $langle 2,x rangle$ is not a principal ideal in $mathbb Z [x]$).
Questions:
1. Are the axes of a torsion-free module over an integral domain isomorphic to each other?
2. Are the axes of a free module over an integral domain lines?
3. If an axis $A$ of a free module over an integral domain is a line, is there a direct sum decomposition $M = A oplus M'$?
abstract-algebra modules direct-sum free-modules
asked Jan 9 at 6:56
Alex CAlex C
10718
$endgroup$
$begingroup$
What do you call a "line" in the second question?
$endgroup$
– lisyarus
Feb 10 at 23:06
$begingroup$
@lisyarus: By line I mean a free module of rank 1.
$endgroup$
– Alex C
Feb 10 at 23:10
add a comment |
$begingroup$
Elements $a$ and $b$ from an $R$-module $M$ are linearly dependent if there are scalars $x$, $y$ in $R$, $x neq 0$ or $y neq 0$, such that $xa = yb$.
Let $M$ be a torsion-free module over an integral domain $R$.
In this case the binary relation has the following properties:
1. It is an equivalence relation on nonzero elements of $M$;
2. Each equivalence class combined with the zero element is a submodule of $M$.
Let's call the submodule an axis of module $M$.
Clearly, if $M$ is a free module over a principal ideal domain, all the axes are lines isomorphic to $R$ over itself.
But what if $M$ is not a free module or $R$ is not a PID?
There is an example of a torsion-free module over an integral domain where the only axis is the module itself, and it is not a line: $langle 2,x rangle$ over $mathbb{Z}[x]$ (Show that $langle 2,x rangle$ is not a principal ideal in $mathbb Z [x]$).
Questions:
1. Are the axes of a torsion-free module over an integral domain isomorphic to each other?
2. Are the axes of a free module over an integral domain lines?
3. If an axis $A$ of a free module over an integral domain is a line, is there a direct sum decomposition $M = A oplus M'$?
abstract-algebra modules direct-sum free-modules
asked Jan 9 at 6:56
Alex CAlex C
10718
$endgroup$
Elements $a$ and $b$ from an $R$-module $M$ are linearly dependent if there are scalars $x$, $y$ in $R$, $x neq 0$ or $y neq 0$, such that $xa = yb$.
Let $M$ be a torsion-free module over an integral domain $R$.
In this case the binary relation has the following properties:
1. It is an equivalence relation on nonzero elements of $M$;
2. Each equivalence class combined with the zero element is a submodule of $M$.
Let's call the submodule an axis of module $M$.
Clearly, if $M$ is a free module over a principal ideal domain, all the axes are lines isomorphic to $R$ over itself.
But what if $M$ is not a free module or $R$ is not a PID?
There is an example of a torsion-free module over an integral domain where the only axis is the module itself, and it is not a line: $langle 2,x rangle$ over $mathbb{Z}[x]$ (Show that $langle 2,x rangle$ is not a principal ideal in $mathbb Z [x]$).
Questions:
1. Are the axes of a torsion-free module over an integral domain isomorphic to each other?
2. Are the axes of a free module over an integral domain lines?
3. If an axis $A$ of a free module over an integral domain is a line, is there a direct sum decomposition $M = A oplus M'$?
abstract-algebra modules direct-sum free-modules
abstract-algebra modules direct-sum free-modules
asked Jan 9 at 6:56
Alex CAlex C
10718
asked Jan 9 at 6:56
Alex CAlex C
10718
edited Feb 22 at 10:53
Alex C
asked Jan 9 at 6:56
Alex CAlex C
10718
asked Jan 9 at 6:56
Alex CAlex C
10718
asked Jan 9 at 6:56
Alex CAlex C
10718
10718
$begingroup$
What do you call a "line" in the second question?
$endgroup$
– lisyarus
Feb 10 at 23:06
$begingroup$
@lisyarus: By line I mean a free module of rank 1.
$endgroup$
– Alex C
Feb 10 at 23:10
add a comment |
$begingroup$
What do you call a "line" in the second question?
$endgroup$
– lisyarus
Feb 10 at 23:06
$begingroup$
@lisyarus: By line I mean a free module of rank 1.
$endgroup$
– Alex C
Feb 10 at 23:10
$begingroup$
What do you call a "line" in the second question?
$endgroup$
– lisyarus
Feb 10 at 23:06
$begingroup$
What do you call a "line" in the second question?
$endgroup$
– lisyarus
Feb 10 at 23:06
$begingroup$
@lisyarus: By line I mean a free module of rank 1.
$endgroup$
– Alex C
Feb 10 at 23:10
$begingroup$
@lisyarus: By line I mean a free module of rank 1.
$endgroup$
– Alex C
Feb 10 at 23:10
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It looks like the answer to the first question is negative:
in $mathbb{Z} times mathbb{R}$ over $mathbb{Z}$ axis $mathbb{Z}$ is not isomorphic to axis $mathbb{R}$.
Trying to find a counterexample for the second question.
answered Jan 11 at 23:06
Alex CAlex C
10718
$endgroup$
$begingroup$
$mathbb R$ is not an axis: e.g. $sqrt 2$ and $sqrt 3$ are linearly independent over $mathbb Z$.
$endgroup$
– lisyarus
Feb 10 at 23:01
1
$begingroup$
On the other hand, in $mathbb Z times mathbb Q$ the $mathbb Q$ part is an axis, providing a counterexample.
$endgroup$
– lisyarus
Feb 10 at 23:06
$begingroup$
@lisyarus: Yes, you are right. Thank you!
$endgroup$
– Alex C
Feb 10 at 23:13
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
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oldest
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oldest
votes
$begingroup$
It looks like the answer to the first question is negative:
in $mathbb{Z} times mathbb{R}$ over $mathbb{Z}$ axis $mathbb{Z}$ is not isomorphic to axis $mathbb{R}$.
Trying to find a counterexample for the second question.
answered Jan 11 at 23:06
Alex CAlex C
10718
$endgroup$
$begingroup$
$mathbb R$ is not an axis: e.g. $sqrt 2$ and $sqrt 3$ are linearly independent over $mathbb Z$.
$endgroup$
– lisyarus
Feb 10 at 23:01
1
$begingroup$
On the other hand, in $mathbb Z times mathbb Q$ the $mathbb Q$ part is an axis, providing a counterexample.
$endgroup$
– lisyarus
Feb 10 at 23:06
$begingroup$
@lisyarus: Yes, you are right. Thank you!
$endgroup$
– Alex C
Feb 10 at 23:13
add a comment |
$begingroup$
It looks like the answer to the first question is negative:
in $mathbb{Z} times mathbb{R}$ over $mathbb{Z}$ axis $mathbb{Z}$ is not isomorphic to axis $mathbb{R}$.
Trying to find a counterexample for the second question.
answered Jan 11 at 23:06
Alex CAlex C
10718
$endgroup$
$begingroup$
$mathbb R$ is not an axis: e.g. $sqrt 2$ and $sqrt 3$ are linearly independent over $mathbb Z$.
$endgroup$
– lisyarus
Feb 10 at 23:01
1
$begingroup$
On the other hand, in $mathbb Z times mathbb Q$ the $mathbb Q$ part is an axis, providing a counterexample.
$endgroup$
– lisyarus
Feb 10 at 23:06
$begingroup$
@lisyarus: Yes, you are right. Thank you!
$endgroup$
– Alex C
Feb 10 at 23:13
add a comment |
$begingroup$
It looks like the answer to the first question is negative:
in $mathbb{Z} times mathbb{R}$ over $mathbb{Z}$ axis $mathbb{Z}$ is not isomorphic to axis $mathbb{R}$.
Trying to find a counterexample for the second question.
answered Jan 11 at 23:06
Alex CAlex C
10718
$endgroup$
It looks like the answer to the first question is negative:
in $mathbb{Z} times mathbb{R}$ over $mathbb{Z}$ axis $mathbb{Z}$ is not isomorphic to axis $mathbb{R}$.
Trying to find a counterexample for the second question.
answered Jan 11 at 23:06
Alex CAlex C
10718
answered Jan 11 at 23:06
Alex CAlex C
10718
answered Jan 11 at 23:06
Alex CAlex C
10718
answered Jan 11 at 23:06
Alex CAlex C
10718
10718
$begingroup$
$mathbb R$ is not an axis: e.g. $sqrt 2$ and $sqrt 3$ are linearly independent over $mathbb Z$.
$endgroup$
– lisyarus
Feb 10 at 23:01
1
$begingroup$
On the other hand, in $mathbb Z times mathbb Q$ the $mathbb Q$ part is an axis, providing a counterexample.
$endgroup$
– lisyarus
Feb 10 at 23:06
$begingroup$
@lisyarus: Yes, you are right. Thank you!
$endgroup$
– Alex C
Feb 10 at 23:13
add a comment |
$begingroup$
$mathbb R$ is not an axis: e.g. $sqrt 2$ and $sqrt 3$ are linearly independent over $mathbb Z$.
$endgroup$
– lisyarus
Feb 10 at 23:01
1
$begingroup$
On the other hand, in $mathbb Z times mathbb Q$ the $mathbb Q$ part is an axis, providing a counterexample.
$endgroup$
– lisyarus
Feb 10 at 23:06
$begingroup$
@lisyarus: Yes, you are right. Thank you!
$endgroup$
– Alex C
Feb 10 at 23:13
$begingroup$
$mathbb R$ is not an axis: e.g. $sqrt 2$ and $sqrt 3$ are linearly independent over $mathbb Z$.
$endgroup$
– lisyarus
Feb 10 at 23:01
$begingroup$
$mathbb R$ is not an axis: e.g. $sqrt 2$ and $sqrt 3$ are linearly independent over $mathbb Z$.
$endgroup$
– lisyarus
Feb 10 at 23:01
1
1
$begingroup$
On the other hand, in $mathbb Z times mathbb Q$ the $mathbb Q$ part is an axis, providing a counterexample.
$endgroup$
– lisyarus
Feb 10 at 23:06
$begingroup$
On the other hand, in $mathbb Z times mathbb Q$ the $mathbb Q$ part is an axis, providing a counterexample.
$endgroup$
– lisyarus
Feb 10 at 23:06
$begingroup$
@lisyarus: Yes, you are right. Thank you!
$endgroup$
– Alex C
Feb 10 at 23:13
$begingroup$
@lisyarus: Yes, you are right. Thank you!
$endgroup$
– Alex C
Feb 10 at 23:13
add a comment |
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$begingroup$
What do you call a "line" in the second question?
$endgroup$
– lisyarus
Feb 10 at 23:06
$begingroup$
@lisyarus: By line I mean a free module of rank 1.
$endgroup$
– Alex C
Feb 10 at 23:10