$A oplus B$ a direct sum of $R$-modules. Possible to have a submodule $X$ of $A oplus B$ s.t. $X cap A = 0 =...
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Let $A oplus B$ be a direct sum of left $R$-modules. Is it possible to have a non-zero submodule $X$ of $A oplus B$ such that $X cap A = 0 = X cap B$?
Perhaps not. A submodule of $A oplus B$ would be of the form $C oplus D$ with $C leq A$, $B leq D$, and both $C$ and $D$ would contain zero so wouldn't $C oplus {0} subset C oplus D cap A$?
abstract-algebra modules
asked Nov 18 at 14:43
Michael Vaughan
751111
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up vote
0
down vote
favorite
Let $A oplus B$ be a direct sum of left $R$-modules. Is it possible to have a non-zero submodule $X$ of $A oplus B$ such that $X cap A = 0 = X cap B$?
Perhaps not. A submodule of $A oplus B$ would be of the form $C oplus D$ with $C leq A$, $B leq D$, and both $C$ and $D$ would contain zero so wouldn't $C oplus {0} subset C oplus D cap A$?
abstract-algebra modules
asked Nov 18 at 14:43
Michael Vaughan
751111
3
Consider the case where A, B, and R are all the real numbers.
– Matthew Towers
Nov 18 at 14:50
Hm, still can't think of anything, lol.
– Michael Vaughan
Nov 18 at 16:53
Well, in that case submodules are the same as subspaces. Does $mathbb{R}^2$ have subspaces that intersect trivially with the coordinate axes?
– Matthew Towers
Nov 18 at 18:20
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $A oplus B$ be a direct sum of left $R$-modules. Is it possible to have a non-zero submodule $X$ of $A oplus B$ such that $X cap A = 0 = X cap B$?
Perhaps not. A submodule of $A oplus B$ would be of the form $C oplus D$ with $C leq A$, $B leq D$, and both $C$ and $D$ would contain zero so wouldn't $C oplus {0} subset C oplus D cap A$?
abstract-algebra modules
asked Nov 18 at 14:43
Michael Vaughan
751111
Let $A oplus B$ be a direct sum of left $R$-modules. Is it possible to have a non-zero submodule $X$ of $A oplus B$ such that $X cap A = 0 = X cap B$?
Perhaps not. A submodule of $A oplus B$ would be of the form $C oplus D$ with $C leq A$, $B leq D$, and both $C$ and $D$ would contain zero so wouldn't $C oplus {0} subset C oplus D cap A$?
abstract-algebra modules
abstract-algebra modules
asked Nov 18 at 14:43
Michael Vaughan
751111
asked Nov 18 at 14:43
Michael Vaughan
751111
asked Nov 18 at 14:43
Michael Vaughan
751111
asked Nov 18 at 14:43
Michael Vaughan
751111
asked Nov 18 at 14:43
Michael Vaughan
751111
751111
3
Consider the case where A, B, and R are all the real numbers.
– Matthew Towers
Nov 18 at 14:50
Hm, still can't think of anything, lol.
– Michael Vaughan
Nov 18 at 16:53
Well, in that case submodules are the same as subspaces. Does $mathbb{R}^2$ have subspaces that intersect trivially with the coordinate axes?
– Matthew Towers
Nov 18 at 18:20
add a comment |
3
Consider the case where A, B, and R are all the real numbers.
– Matthew Towers
Nov 18 at 14:50
Hm, still can't think of anything, lol.
– Michael Vaughan
Nov 18 at 16:53
Well, in that case submodules are the same as subspaces. Does $mathbb{R}^2$ have subspaces that intersect trivially with the coordinate axes?
– Matthew Towers
Nov 18 at 18:20
3
3
Consider the case where A, B, and R are all the real numbers.
– Matthew Towers
Nov 18 at 14:50
Consider the case where A, B, and R are all the real numbers.
– Matthew Towers
Nov 18 at 14:50
Hm, still can't think of anything, lol.
– Michael Vaughan
Nov 18 at 16:53
Hm, still can't think of anything, lol.
– Michael Vaughan
Nov 18 at 16:53
Well, in that case submodules are the same as subspaces. Does $mathbb{R}^2$ have subspaces that intersect trivially with the coordinate axes?
– Matthew Towers
Nov 18 at 18:20
Well, in that case submodules are the same as subspaces. Does $mathbb{R}^2$ have subspaces that intersect trivially with the coordinate axes?
– Matthew Towers
Nov 18 at 18:20
add a comment |
1 Answer
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It depends on $A$ and $B$ (and $R$).
For example, if $R$ is a field and $A,B$ are nonzero $R$-vector spaces, then there are ($1$-dimensional) vector subspaces of $Aoplus B$ not contained in $A$ or $B$.
On the other hand, if $R=mathbb{Z}$ and $A=mathbb{Z}/2mathbb{Z}$, $B=mathbb{Z}/3mathbb{Z}$, then the only nontrivial submodules of $Aoplus B=mathbb{Z}/6mathbb{Z}$ are $A,B$ and $Aoplus B$.
answered Nov 18 at 14:59
user10354138
6,8051623
I realize that containment wouldn't hold, but would the intersection be trivial?
– Michael Vaughan
Nov 18 at 15:17
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
It depends on $A$ and $B$ (and $R$).
For example, if $R$ is a field and $A,B$ are nonzero $R$-vector spaces, then there are ($1$-dimensional) vector subspaces of $Aoplus B$ not contained in $A$ or $B$.
On the other hand, if $R=mathbb{Z}$ and $A=mathbb{Z}/2mathbb{Z}$, $B=mathbb{Z}/3mathbb{Z}$, then the only nontrivial submodules of $Aoplus B=mathbb{Z}/6mathbb{Z}$ are $A,B$ and $Aoplus B$.
answered Nov 18 at 14:59
user10354138
6,8051623
I realize that containment wouldn't hold, but would the intersection be trivial?
– Michael Vaughan
Nov 18 at 15:17
add a comment |
up vote
0
down vote
It depends on $A$ and $B$ (and $R$).
For example, if $R$ is a field and $A,B$ are nonzero $R$-vector spaces, then there are ($1$-dimensional) vector subspaces of $Aoplus B$ not contained in $A$ or $B$.
On the other hand, if $R=mathbb{Z}$ and $A=mathbb{Z}/2mathbb{Z}$, $B=mathbb{Z}/3mathbb{Z}$, then the only nontrivial submodules of $Aoplus B=mathbb{Z}/6mathbb{Z}$ are $A,B$ and $Aoplus B$.
answered Nov 18 at 14:59
user10354138
6,8051623
I realize that containment wouldn't hold, but would the intersection be trivial?
– Michael Vaughan
Nov 18 at 15:17
add a comment |
up vote
0
down vote
up vote
0
down vote
It depends on $A$ and $B$ (and $R$).
For example, if $R$ is a field and $A,B$ are nonzero $R$-vector spaces, then there are ($1$-dimensional) vector subspaces of $Aoplus B$ not contained in $A$ or $B$.
On the other hand, if $R=mathbb{Z}$ and $A=mathbb{Z}/2mathbb{Z}$, $B=mathbb{Z}/3mathbb{Z}$, then the only nontrivial submodules of $Aoplus B=mathbb{Z}/6mathbb{Z}$ are $A,B$ and $Aoplus B$.
answered Nov 18 at 14:59
user10354138
6,8051623
It depends on $A$ and $B$ (and $R$).
For example, if $R$ is a field and $A,B$ are nonzero $R$-vector spaces, then there are ($1$-dimensional) vector subspaces of $Aoplus B$ not contained in $A$ or $B$.
On the other hand, if $R=mathbb{Z}$ and $A=mathbb{Z}/2mathbb{Z}$, $B=mathbb{Z}/3mathbb{Z}$, then the only nontrivial submodules of $Aoplus B=mathbb{Z}/6mathbb{Z}$ are $A,B$ and $Aoplus B$.
answered Nov 18 at 14:59
user10354138
6,8051623
edited Nov 18 at 15:18
answered Nov 18 at 14:59
user10354138
6,8051623
answered Nov 18 at 14:59
user10354138
6,8051623
answered Nov 18 at 14:59
user10354138
6,8051623
6,8051623
I realize that containment wouldn't hold, but would the intersection be trivial?
– Michael Vaughan
Nov 18 at 15:17
add a comment |
I realize that containment wouldn't hold, but would the intersection be trivial?
– Michael Vaughan
Nov 18 at 15:17
I realize that containment wouldn't hold, but would the intersection be trivial?
– Michael Vaughan
Nov 18 at 15:17
I realize that containment wouldn't hold, but would the intersection be trivial?
– Michael Vaughan
Nov 18 at 15:17
add a comment |
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3
Consider the case where A, B, and R are all the real numbers.
– Matthew Towers
Nov 18 at 14:50
Hm, still can't think of anything, lol.
– Michael Vaughan
Nov 18 at 16:53
Well, in that case submodules are the same as subspaces. Does $mathbb{R}^2$ have subspaces that intersect trivially with the coordinate axes?
– Matthew Towers
Nov 18 at 18:20