If $M$ is a finitely generated torsion module over a PID which is not a field then $M$ is Artinian.
up vote
1
down vote
favorite
If $M$ is a finitely generated torsion module over a PID which is not a field then $M$ is Artinian.
Is contradiction the way to go? How does $M$ being torsion help us? Any hints on how to prove?
linear-algebra
add a comment |
up vote
1
down vote
favorite
If $M$ is a finitely generated torsion module over a PID which is not a field then $M$ is Artinian.
Is contradiction the way to go? How does $M$ being torsion help us? Any hints on how to prove?
linear-algebra
@Youngsu Well, $mathbb Z/2mathbb Ztimes mathbb Z/2mathbb Z$ is a counterexample to that... but nevertheless, using the structure theorem is the right way to go...
– rschwieb
Nov 22 at 19:48
You are absolutely right. I should've said product of those. Thank you for pointing it out.
– Youngsu
Nov 22 at 20:24
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
If $M$ is a finitely generated torsion module over a PID which is not a field then $M$ is Artinian.
Is contradiction the way to go? How does $M$ being torsion help us? Any hints on how to prove?
linear-algebra
If $M$ is a finitely generated torsion module over a PID which is not a field then $M$ is Artinian.
Is contradiction the way to go? How does $M$ being torsion help us? Any hints on how to prove?
linear-algebra
linear-algebra
edited Nov 22 at 20:00
Bernard
117k637109
117k637109
asked Nov 22 at 19:15
UserA
496216
496216
@Youngsu Well, $mathbb Z/2mathbb Ztimes mathbb Z/2mathbb Z$ is a counterexample to that... but nevertheless, using the structure theorem is the right way to go...
– rschwieb
Nov 22 at 19:48
You are absolutely right. I should've said product of those. Thank you for pointing it out.
– Youngsu
Nov 22 at 20:24
add a comment |
@Youngsu Well, $mathbb Z/2mathbb Ztimes mathbb Z/2mathbb Z$ is a counterexample to that... but nevertheless, using the structure theorem is the right way to go...
– rschwieb
Nov 22 at 19:48
You are absolutely right. I should've said product of those. Thank you for pointing it out.
– Youngsu
Nov 22 at 20:24
@Youngsu Well, $mathbb Z/2mathbb Ztimes mathbb Z/2mathbb Z$ is a counterexample to that... but nevertheless, using the structure theorem is the right way to go...
– rschwieb
Nov 22 at 19:48
@Youngsu Well, $mathbb Z/2mathbb Ztimes mathbb Z/2mathbb Z$ is a counterexample to that... but nevertheless, using the structure theorem is the right way to go...
– rschwieb
Nov 22 at 19:48
You are absolutely right. I should've said product of those. Thank you for pointing it out.
– Youngsu
Nov 22 at 20:24
You are absolutely right. I should've said product of those. Thank you for pointing it out.
– Youngsu
Nov 22 at 20:24
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
accepted
How does M being torsion help us?
I assume you are learning about the fundamental structure theorem on f.g. modules over a PID?
The fact that the module is f.g. and torsion implies $Mcong oplus_{i=1}^n R/(p_i^{e_i})$ for some primes $p_i$ in $R$ and exponents $e_i$.
Any hints on how to prove?
You can check that each of these factors has finite composition length, and so the finite product of the factors has finite composition length.
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3009531%2fif-m-is-a-finitely-generated-torsion-module-over-a-pid-which-is-not-a-field-th%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
How does M being torsion help us?
I assume you are learning about the fundamental structure theorem on f.g. modules over a PID?
The fact that the module is f.g. and torsion implies $Mcong oplus_{i=1}^n R/(p_i^{e_i})$ for some primes $p_i$ in $R$ and exponents $e_i$.
Any hints on how to prove?
You can check that each of these factors has finite composition length, and so the finite product of the factors has finite composition length.
add a comment |
up vote
0
down vote
accepted
How does M being torsion help us?
I assume you are learning about the fundamental structure theorem on f.g. modules over a PID?
The fact that the module is f.g. and torsion implies $Mcong oplus_{i=1}^n R/(p_i^{e_i})$ for some primes $p_i$ in $R$ and exponents $e_i$.
Any hints on how to prove?
You can check that each of these factors has finite composition length, and so the finite product of the factors has finite composition length.
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
How does M being torsion help us?
I assume you are learning about the fundamental structure theorem on f.g. modules over a PID?
The fact that the module is f.g. and torsion implies $Mcong oplus_{i=1}^n R/(p_i^{e_i})$ for some primes $p_i$ in $R$ and exponents $e_i$.
Any hints on how to prove?
You can check that each of these factors has finite composition length, and so the finite product of the factors has finite composition length.
How does M being torsion help us?
I assume you are learning about the fundamental structure theorem on f.g. modules over a PID?
The fact that the module is f.g. and torsion implies $Mcong oplus_{i=1}^n R/(p_i^{e_i})$ for some primes $p_i$ in $R$ and exponents $e_i$.
Any hints on how to prove?
You can check that each of these factors has finite composition length, and so the finite product of the factors has finite composition length.
answered Nov 22 at 19:53
rschwieb
104k1299241
104k1299241
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3009531%2fif-m-is-a-finitely-generated-torsion-module-over-a-pid-which-is-not-a-field-th%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
@Youngsu Well, $mathbb Z/2mathbb Ztimes mathbb Z/2mathbb Z$ is a counterexample to that... but nevertheless, using the structure theorem is the right way to go...
– rschwieb
Nov 22 at 19:48
You are absolutely right. I should've said product of those. Thank you for pointing it out.
– Youngsu
Nov 22 at 20:24