Right ideals in a matrix ring
Are there right ideals of $M_n(k)$ other than those of the form of zeroing one or more rows at a time?
For example you can take first row to be zero and this is a right ideal. My question is are there ideals other than those of this form?
abstract-algebra matrices ring-theory ideals
edited Dec 27 '18 at 22:16
user26857
39.3k123983
asked Jun 26 '17 at 16:00
Ronit Debnath
546115
add a comment |
Are there right ideals of $M_n(k)$ other than those of the form of zeroing one or more rows at a time?
For example you can take first row to be zero and this is a right ideal. My question is are there ideals other than those of this form?
abstract-algebra matrices ring-theory ideals
edited Dec 27 '18 at 22:16
user26857
39.3k123983
asked Jun 26 '17 at 16:00
Ronit Debnath
546115
You just have to pick something likely and check. It isn't hard at all to find an example that way.
– rschwieb
Jun 26 '17 at 16:22
A generalization of this example would be: let $W$ be a linear subspace of $k^n$. Then the matrices of $M_n(k)$ such that every column is in $W$ is a right ideal of $M_n(k)$. (So in rschwieb's example, $W$ is the span of $(1, 1)$. And in your example, $W$ is the subspace of vectors with some entries fixed to 0.)
– Daniel Schepler
Jun 26 '17 at 16:42
In fact, I've convinced myself every right ideal of $M_n(k)$ is of this form for some subspace $W$ - and the proof is such that it could reasonably be a problem in an upper-division undergraduate course. If you're interested, you could try to work out a proof, and then post another question if you run into trouble.
– Daniel Schepler
Jun 26 '17 at 16:56
add a comment |
Are there right ideals of $M_n(k)$ other than those of the form of zeroing one or more rows at a time?
For example you can take first row to be zero and this is a right ideal. My question is are there ideals other than those of this form?
abstract-algebra matrices ring-theory ideals
edited Dec 27 '18 at 22:16
user26857
39.3k123983
asked Jun 26 '17 at 16:00
Ronit Debnath
546115
Are there right ideals of $M_n(k)$ other than those of the form of zeroing one or more rows at a time?
For example you can take first row to be zero and this is a right ideal. My question is are there ideals other than those of this form?
abstract-algebra matrices ring-theory ideals
abstract-algebra matrices ring-theory ideals
edited Dec 27 '18 at 22:16
user26857
39.3k123983
asked Jun 26 '17 at 16:00
Ronit Debnath
546115
edited Dec 27 '18 at 22:16
user26857
39.3k123983
asked Jun 26 '17 at 16:00
Ronit Debnath
546115
edited Dec 27 '18 at 22:16
user26857
39.3k123983
edited Dec 27 '18 at 22:16
user26857
39.3k123983
edited Dec 27 '18 at 22:16
user26857
39.3k123983
39.3k123983
asked Jun 26 '17 at 16:00
Ronit Debnath
546115
asked Jun 26 '17 at 16:00
Ronit Debnath
546115
asked Jun 26 '17 at 16:00
Ronit Debnath
546115
546115
You just have to pick something likely and check. It isn't hard at all to find an example that way.
– rschwieb
Jun 26 '17 at 16:22
A generalization of this example would be: let $W$ be a linear subspace of $k^n$. Then the matrices of $M_n(k)$ such that every column is in $W$ is a right ideal of $M_n(k)$. (So in rschwieb's example, $W$ is the span of $(1, 1)$. And in your example, $W$ is the subspace of vectors with some entries fixed to 0.)
– Daniel Schepler
Jun 26 '17 at 16:42
In fact, I've convinced myself every right ideal of $M_n(k)$ is of this form for some subspace $W$ - and the proof is such that it could reasonably be a problem in an upper-division undergraduate course. If you're interested, you could try to work out a proof, and then post another question if you run into trouble.
– Daniel Schepler
Jun 26 '17 at 16:56
add a comment |
You just have to pick something likely and check. It isn't hard at all to find an example that way.
– rschwieb
Jun 26 '17 at 16:22
A generalization of this example would be: let $W$ be a linear subspace of $k^n$. Then the matrices of $M_n(k)$ such that every column is in $W$ is a right ideal of $M_n(k)$. (So in rschwieb's example, $W$ is the span of $(1, 1)$. And in your example, $W$ is the subspace of vectors with some entries fixed to 0.)
– Daniel Schepler
Jun 26 '17 at 16:42
In fact, I've convinced myself every right ideal of $M_n(k)$ is of this form for some subspace $W$ - and the proof is such that it could reasonably be a problem in an upper-division undergraduate course. If you're interested, you could try to work out a proof, and then post another question if you run into trouble.
– Daniel Schepler
Jun 26 '17 at 16:56
You just have to pick something likely and check. It isn't hard at all to find an example that way.
– rschwieb
Jun 26 '17 at 16:22
You just have to pick something likely and check. It isn't hard at all to find an example that way.
– rschwieb
Jun 26 '17 at 16:22
A generalization of this example would be: let $W$ be a linear subspace of $k^n$. Then the matrices of $M_n(k)$ such that every column is in $W$ is a right ideal of $M_n(k)$. (So in rschwieb's example, $W$ is the span of $(1, 1)$. And in your example, $W$ is the subspace of vectors with some entries fixed to 0.)
– Daniel Schepler
Jun 26 '17 at 16:42
A generalization of this example would be: let $W$ be a linear subspace of $k^n$. Then the matrices of $M_n(k)$ such that every column is in $W$ is a right ideal of $M_n(k)$. (So in rschwieb's example, $W$ is the span of $(1, 1)$. And in your example, $W$ is the subspace of vectors with some entries fixed to 0.)
– Daniel Schepler
Jun 26 '17 at 16:42
In fact, I've convinced myself every right ideal of $M_n(k)$ is of this form for some subspace $W$ - and the proof is such that it could reasonably be a problem in an upper-division undergraduate course. If you're interested, you could try to work out a proof, and then post another question if you run into trouble.
– Daniel Schepler
Jun 26 '17 at 16:56
In fact, I've convinced myself every right ideal of $M_n(k)$ is of this form for some subspace $W$ - and the proof is such that it could reasonably be a problem in an upper-division undergraduate course. If you're interested, you could try to work out a proof, and then post another question if you run into trouble.
– Daniel Schepler
Jun 26 '17 at 16:56
add a comment |
2 Answers
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Yes, for example
$$begin{bmatrix}1&1\1&1end{bmatrix}M_2(k)=left{begin{bmatrix}a&b\a&b end{bmatrix}middle|,a,bin kright}$$
It has nonzero entries in all rows, but it is clearly not the entire ring.
Another counting argument that works for infinite $k$: There are only $2^n$ possibilities for the ideals you're describing, but there are infinitely many right ideals if $k$ is infinite. (Even if $k$ is finite the example above shows that finiteness does not help.)
answered Jun 26 '17 at 16:17
rschwieb
105k1299244
add a comment |
As a generalization of your example, let $W$ be a linear subspace of $k^n$. Then the set of $n times n$ matrices such that every column is in $W$, or equivalently, ${ A in M_n(k) , | , CS(A) subseteq W }$, is a right ideal of $M_n(k)$. (The additivity, and containing 0, are fairly easy; for the right multiplication condition, use the fact that $CS(AB) subseteq CS(A)$.)
In your example, $W$ is the subspace of vectors with the corresponding elements of the vector equal to 0. And in rschwieb's example, $W$ is the span of $(1, 1)$.
In fact, it turns out that every right ideal of $M_n(k)$ is of this form for some subspace $W$. The proof isn't conceptually hard (though it might possibly be cumbersome notation-wise).
answered Jun 26 '17 at 17:13
Daniel Schepler
8,2841618
add a comment |
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2 Answers
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2 Answers
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Yes, for example
$$begin{bmatrix}1&1\1&1end{bmatrix}M_2(k)=left{begin{bmatrix}a&b\a&b end{bmatrix}middle|,a,bin kright}$$
It has nonzero entries in all rows, but it is clearly not the entire ring.
Another counting argument that works for infinite $k$: There are only $2^n$ possibilities for the ideals you're describing, but there are infinitely many right ideals if $k$ is infinite. (Even if $k$ is finite the example above shows that finiteness does not help.)
answered Jun 26 '17 at 16:17
rschwieb
105k1299244
add a comment |
Yes, for example
$$begin{bmatrix}1&1\1&1end{bmatrix}M_2(k)=left{begin{bmatrix}a&b\a&b end{bmatrix}middle|,a,bin kright}$$
It has nonzero entries in all rows, but it is clearly not the entire ring.
Another counting argument that works for infinite $k$: There are only $2^n$ possibilities for the ideals you're describing, but there are infinitely many right ideals if $k$ is infinite. (Even if $k$ is finite the example above shows that finiteness does not help.)
answered Jun 26 '17 at 16:17
rschwieb
105k1299244
add a comment |
Yes, for example
$$begin{bmatrix}1&1\1&1end{bmatrix}M_2(k)=left{begin{bmatrix}a&b\a&b end{bmatrix}middle|,a,bin kright}$$
It has nonzero entries in all rows, but it is clearly not the entire ring.
Another counting argument that works for infinite $k$: There are only $2^n$ possibilities for the ideals you're describing, but there are infinitely many right ideals if $k$ is infinite. (Even if $k$ is finite the example above shows that finiteness does not help.)
answered Jun 26 '17 at 16:17
rschwieb
105k1299244
Yes, for example
$$begin{bmatrix}1&1\1&1end{bmatrix}M_2(k)=left{begin{bmatrix}a&b\a&b end{bmatrix}middle|,a,bin kright}$$
It has nonzero entries in all rows, but it is clearly not the entire ring.
Another counting argument that works for infinite $k$: There are only $2^n$ possibilities for the ideals you're describing, but there are infinitely many right ideals if $k$ is infinite. (Even if $k$ is finite the example above shows that finiteness does not help.)
answered Jun 26 '17 at 16:17
rschwieb
105k1299244
edited Jun 26 '17 at 16:23
answered Jun 26 '17 at 16:17
rschwieb
105k1299244
answered Jun 26 '17 at 16:17
rschwieb
105k1299244
answered Jun 26 '17 at 16:17
rschwieb
105k1299244
105k1299244
add a comment |
add a comment |
As a generalization of your example, let $W$ be a linear subspace of $k^n$. Then the set of $n times n$ matrices such that every column is in $W$, or equivalently, ${ A in M_n(k) , | , CS(A) subseteq W }$, is a right ideal of $M_n(k)$. (The additivity, and containing 0, are fairly easy; for the right multiplication condition, use the fact that $CS(AB) subseteq CS(A)$.)
In your example, $W$ is the subspace of vectors with the corresponding elements of the vector equal to 0. And in rschwieb's example, $W$ is the span of $(1, 1)$.
In fact, it turns out that every right ideal of $M_n(k)$ is of this form for some subspace $W$. The proof isn't conceptually hard (though it might possibly be cumbersome notation-wise).
answered Jun 26 '17 at 17:13
Daniel Schepler
8,2841618
add a comment |
As a generalization of your example, let $W$ be a linear subspace of $k^n$. Then the set of $n times n$ matrices such that every column is in $W$, or equivalently, ${ A in M_n(k) , | , CS(A) subseteq W }$, is a right ideal of $M_n(k)$. (The additivity, and containing 0, are fairly easy; for the right multiplication condition, use the fact that $CS(AB) subseteq CS(A)$.)
In your example, $W$ is the subspace of vectors with the corresponding elements of the vector equal to 0. And in rschwieb's example, $W$ is the span of $(1, 1)$.
In fact, it turns out that every right ideal of $M_n(k)$ is of this form for some subspace $W$. The proof isn't conceptually hard (though it might possibly be cumbersome notation-wise).
answered Jun 26 '17 at 17:13
Daniel Schepler
8,2841618
add a comment |
As a generalization of your example, let $W$ be a linear subspace of $k^n$. Then the set of $n times n$ matrices such that every column is in $W$, or equivalently, ${ A in M_n(k) , | , CS(A) subseteq W }$, is a right ideal of $M_n(k)$. (The additivity, and containing 0, are fairly easy; for the right multiplication condition, use the fact that $CS(AB) subseteq CS(A)$.)
In your example, $W$ is the subspace of vectors with the corresponding elements of the vector equal to 0. And in rschwieb's example, $W$ is the span of $(1, 1)$.
In fact, it turns out that every right ideal of $M_n(k)$ is of this form for some subspace $W$. The proof isn't conceptually hard (though it might possibly be cumbersome notation-wise).
answered Jun 26 '17 at 17:13
Daniel Schepler
8,2841618
As a generalization of your example, let $W$ be a linear subspace of $k^n$. Then the set of $n times n$ matrices such that every column is in $W$, or equivalently, ${ A in M_n(k) , | , CS(A) subseteq W }$, is a right ideal of $M_n(k)$. (The additivity, and containing 0, are fairly easy; for the right multiplication condition, use the fact that $CS(AB) subseteq CS(A)$.)
In your example, $W$ is the subspace of vectors with the corresponding elements of the vector equal to 0. And in rschwieb's example, $W$ is the span of $(1, 1)$.
In fact, it turns out that every right ideal of $M_n(k)$ is of this form for some subspace $W$. The proof isn't conceptually hard (though it might possibly be cumbersome notation-wise).
answered Jun 26 '17 at 17:13
Daniel Schepler
8,2841618
answered Jun 26 '17 at 17:13
Daniel Schepler
8,2841618
answered Jun 26 '17 at 17:13
Daniel Schepler
8,2841618
answered Jun 26 '17 at 17:13
Daniel Schepler
8,2841618
8,2841618
add a comment |
add a comment |
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You just have to pick something likely and check. It isn't hard at all to find an example that way.
– rschwieb
Jun 26 '17 at 16:22
A generalization of this example would be: let $W$ be a linear subspace of $k^n$. Then the matrices of $M_n(k)$ such that every column is in $W$ is a right ideal of $M_n(k)$. (So in rschwieb's example, $W$ is the span of $(1, 1)$. And in your example, $W$ is the subspace of vectors with some entries fixed to 0.)
– Daniel Schepler
Jun 26 '17 at 16:42
In fact, I've convinced myself every right ideal of $M_n(k)$ is of this form for some subspace $W$ - and the proof is such that it could reasonably be a problem in an upper-division undergraduate course. If you're interested, you could try to work out a proof, and then post another question if you run into trouble.
– Daniel Schepler
Jun 26 '17 at 16:56