Mixing
Finitely generated Tensor Product
$begingroup$
Does $M otimes N $ finitely generated imply that $M$ and $N$ are also finitely generated? I know that the converse is true, but I'm not really sure about this one. Thanks.
abstract-algebra modules
edited Jan 22 at 20:43
user26857
39.3k124183
asked Jan 22 at 17:39
Pedro SantosPedro Santos
1539
$endgroup$
add a comment |
$begingroup$
Does $M otimes N $ finitely generated imply that $M$ and $N$ are also finitely generated? I know that the converse is true, but I'm not really sure about this one. Thanks.
abstract-algebra modules
edited Jan 22 at 20:43
user26857
39.3k124183
asked Jan 22 at 17:39
Pedro SantosPedro Santos
1539
$endgroup$
add a comment |
$begingroup$
Does $M otimes N $ finitely generated imply that $M$ and $N$ are also finitely generated? I know that the converse is true, but I'm not really sure about this one. Thanks.
abstract-algebra modules
edited Jan 22 at 20:43
user26857
39.3k124183
asked Jan 22 at 17:39
Pedro SantosPedro Santos
1539
$endgroup$
Does $M otimes N $ finitely generated imply that $M$ and $N$ are also finitely generated? I know that the converse is true, but I'm not really sure about this one. Thanks.
abstract-algebra modules
abstract-algebra modules
edited Jan 22 at 20:43
user26857
39.3k124183
asked Jan 22 at 17:39
Pedro SantosPedro Santos
1539
edited Jan 22 at 20:43
user26857
39.3k124183
asked Jan 22 at 17:39
Pedro SantosPedro Santos
1539
edited Jan 22 at 20:43
user26857
39.3k124183
edited Jan 22 at 20:43
user26857
39.3k124183
edited Jan 22 at 20:43
user26857
39.3k124183
39.3k124183
asked Jan 22 at 17:39
Pedro SantosPedro Santos
1539
asked Jan 22 at 17:39
Pedro SantosPedro Santos
1539
asked Jan 22 at 17:39
Pedro SantosPedro Santos
1539
1539
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
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No : there are examples where neither is finitely generated, and examples where one of them is and not the other: take for instance $mathbb{Q}otimestext{(any torsion abelian group)}$, which is $0$.
Now it's known that there are finitely generated and infinitely generated torsion abelian groups, so that's it.
edited Jan 22 at 20:44
user26857
39.3k124183
answered Jan 22 at 17:43
MaxMax
15.2k11143
$endgroup$
$begingroup$
Oh yeah , Thanks.
$endgroup$
– Pedro Santos
Jan 22 at 17:44
$begingroup$
Can u give me an example of a infinitely generated torsion abelian group? Does $mathbb{Z}_{2}^mathbb{N} work ?$
$endgroup$
– Pedro Santos
Jan 22 at 17:49
$begingroup$
nevermind i understand it now
$endgroup$
– Pedro Santos
Jan 22 at 17:56
add a comment |
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1 Answer
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1 Answer
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$begingroup$
No : there are examples where neither is finitely generated, and examples where one of them is and not the other: take for instance $mathbb{Q}otimestext{(any torsion abelian group)}$, which is $0$.
Now it's known that there are finitely generated and infinitely generated torsion abelian groups, so that's it.
edited Jan 22 at 20:44
user26857
39.3k124183
answered Jan 22 at 17:43
MaxMax
15.2k11143
$endgroup$
$begingroup$
Oh yeah , Thanks.
$endgroup$
– Pedro Santos
Jan 22 at 17:44
$begingroup$
Can u give me an example of a infinitely generated torsion abelian group? Does $mathbb{Z}_{2}^mathbb{N} work ?$
$endgroup$
– Pedro Santos
Jan 22 at 17:49
$begingroup$
nevermind i understand it now
$endgroup$
– Pedro Santos
Jan 22 at 17:56
add a comment |
$begingroup$
No : there are examples where neither is finitely generated, and examples where one of them is and not the other: take for instance $mathbb{Q}otimestext{(any torsion abelian group)}$, which is $0$.
Now it's known that there are finitely generated and infinitely generated torsion abelian groups, so that's it.
edited Jan 22 at 20:44
user26857
39.3k124183
answered Jan 22 at 17:43
MaxMax
15.2k11143
$endgroup$
$begingroup$
Oh yeah , Thanks.
$endgroup$
– Pedro Santos
Jan 22 at 17:44
$begingroup$
Can u give me an example of a infinitely generated torsion abelian group? Does $mathbb{Z}_{2}^mathbb{N} work ?$
$endgroup$
– Pedro Santos
Jan 22 at 17:49
$begingroup$
nevermind i understand it now
$endgroup$
– Pedro Santos
Jan 22 at 17:56
add a comment |
$begingroup$
No : there are examples where neither is finitely generated, and examples where one of them is and not the other: take for instance $mathbb{Q}otimestext{(any torsion abelian group)}$, which is $0$.
Now it's known that there are finitely generated and infinitely generated torsion abelian groups, so that's it.
edited Jan 22 at 20:44
user26857
39.3k124183
answered Jan 22 at 17:43
MaxMax
15.2k11143
$endgroup$
No : there are examples where neither is finitely generated, and examples where one of them is and not the other: take for instance $mathbb{Q}otimestext{(any torsion abelian group)}$, which is $0$.
Now it's known that there are finitely generated and infinitely generated torsion abelian groups, so that's it.
edited Jan 22 at 20:44
user26857
39.3k124183
answered Jan 22 at 17:43
MaxMax
15.2k11143
edited Jan 22 at 20:44
user26857
39.3k124183
edited Jan 22 at 20:44
user26857
39.3k124183
edited Jan 22 at 20:44
user26857
39.3k124183
39.3k124183
answered Jan 22 at 17:43
MaxMax
15.2k11143
answered Jan 22 at 17:43
MaxMax
15.2k11143
answered Jan 22 at 17:43
MaxMax
15.2k11143
15.2k11143
$begingroup$
Oh yeah , Thanks.
$endgroup$
– Pedro Santos
Jan 22 at 17:44
$begingroup$
Can u give me an example of a infinitely generated torsion abelian group? Does $mathbb{Z}_{2}^mathbb{N} work ?$
$endgroup$
– Pedro Santos
Jan 22 at 17:49
$begingroup$
nevermind i understand it now
$endgroup$
– Pedro Santos
Jan 22 at 17:56
add a comment |
$begingroup$
Oh yeah , Thanks.
$endgroup$
– Pedro Santos
Jan 22 at 17:44
$begingroup$
Can u give me an example of a infinitely generated torsion abelian group? Does $mathbb{Z}_{2}^mathbb{N} work ?$
$endgroup$
– Pedro Santos
Jan 22 at 17:49
$begingroup$
nevermind i understand it now
$endgroup$
– Pedro Santos
Jan 22 at 17:56
$begingroup$
Oh yeah , Thanks.
$endgroup$
– Pedro Santos
Jan 22 at 17:44
$begingroup$
Oh yeah , Thanks.
$endgroup$
– Pedro Santos
Jan 22 at 17:44
$begingroup$
Can u give me an example of a infinitely generated torsion abelian group? Does $mathbb{Z}_{2}^mathbb{N} work ?$
$endgroup$
– Pedro Santos
Jan 22 at 17:49
$begingroup$
Can u give me an example of a infinitely generated torsion abelian group? Does $mathbb{Z}_{2}^mathbb{N} work ?$
$endgroup$
– Pedro Santos
Jan 22 at 17:49
$begingroup$
nevermind i understand it now
$endgroup$
– Pedro Santos
Jan 22 at 17:56
$begingroup$
nevermind i understand it now
$endgroup$
– Pedro Santos
Jan 22 at 17:56
add a comment |
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