How do you write this module as a direct sum of cyclic modules?
0
$begingroup$
Let M be the module over Z [i] generated by elements x, y whose relations are determined by $(1+i)x+(2-i)y=0$ and $3x+5y=0$. How can one write M as a direct sum of cyclic modules?
abstract-algebra modules
edited Dec 2 '18 at 0:42
rschwieb
105k12101246
asked Dec 1 '18 at 20:23
Mike HawkMike Hawk
61
$endgroup$
|
show 1 more comment
0
$begingroup$
Let M be the module over Z [i] generated by elements x, y whose relations are determined by $(1+i)x+(2-i)y=0$ and $3x+5y=0$. How can one write M as a direct sum of cyclic modules?
abstract-algebra modules
edited Dec 2 '18 at 0:42
rschwieb
105k12101246
asked Dec 1 '18 at 20:23
Mike HawkMike Hawk
61
$endgroup$
1
$begingroup$
Hey, and welcome to M.SE! To help you, it is helpful to know what you have tried, what your background is (e.g., are you comfortable with the notion of direct sum and cyclic module?), and what your process of thinking was when you approached this question.
$endgroup$
– Luke
Dec 1 '18 at 20:42
$begingroup$
@Luke, I'm learning modules for the first time. So that, when I think of a module I'm always with the generalization of a vector space. When I think of direct sum I always think on the linear combination of linearly independent generators of subspaces. But I don't know how can I relate this thinking with that question.
$endgroup$
– Mike Hawk
Dec 1 '18 at 22:29
$begingroup$
@MikeHawk you seem to have confused the title field for the tags field. I changed the title to be more useful. You should do something like that next time. Remember, the goal is to be informative and searchable. If you don’t, well, you’ll probably find out what the consequences are. Good luck.
$endgroup$
– rschwieb
Dec 2 '18 at 0:43
$begingroup$
Thank you @rschwieb
$endgroup$
– Mike Hawk
Dec 2 '18 at 0:56
$begingroup$
Similar to these: 1, 2, 3
$endgroup$
– André 3000
Dec 2 '18 at 8:39
|
show 1 more comment
0
0
0
$begingroup$
Let M be the module over Z [i] generated by elements x, y whose relations are determined by $(1+i)x+(2-i)y=0$ and $3x+5y=0$. How can one write M as a direct sum of cyclic modules?
abstract-algebra modules
edited Dec 2 '18 at 0:42
rschwieb
105k12101246
asked Dec 1 '18 at 20:23
Mike HawkMike Hawk
61
$endgroup$
Let M be the module over Z [i] generated by elements x, y whose relations are determined by $(1+i)x+(2-i)y=0$ and $3x+5y=0$. How can one write M as a direct sum of cyclic modules?
abstract-algebra modules
abstract-algebra modules
edited Dec 2 '18 at 0:42
rschwieb
105k12101246
asked Dec 1 '18 at 20:23
Mike HawkMike Hawk
61
edited Dec 2 '18 at 0:42
rschwieb
105k12101246
asked Dec 1 '18 at 20:23
Mike HawkMike Hawk
61
edited Dec 2 '18 at 0:42
rschwieb
105k12101246
edited Dec 2 '18 at 0:42
rschwieb
105k12101246
edited Dec 2 '18 at 0:42
rschwieb
105k12101246
105k12101246
asked Dec 1 '18 at 20:23
Mike HawkMike Hawk
61
asked Dec 1 '18 at 20:23
Mike HawkMike Hawk
61
asked Dec 1 '18 at 20:23
Mike HawkMike Hawk
61
61
1
$begingroup$
Hey, and welcome to M.SE! To help you, it is helpful to know what you have tried, what your background is (e.g., are you comfortable with the notion of direct sum and cyclic module?), and what your process of thinking was when you approached this question.
$endgroup$
– Luke
Dec 1 '18 at 20:42
$begingroup$
@Luke, I'm learning modules for the first time. So that, when I think of a module I'm always with the generalization of a vector space. When I think of direct sum I always think on the linear combination of linearly independent generators of subspaces. But I don't know how can I relate this thinking with that question.
$endgroup$
– Mike Hawk
Dec 1 '18 at 22:29
$begingroup$
@MikeHawk you seem to have confused the title field for the tags field. I changed the title to be more useful. You should do something like that next time. Remember, the goal is to be informative and searchable. If you don’t, well, you’ll probably find out what the consequences are. Good luck.
$endgroup$
– rschwieb
Dec 2 '18 at 0:43
$begingroup$
Thank you @rschwieb
$endgroup$
– Mike Hawk
Dec 2 '18 at 0:56
$begingroup$
Similar to these: 1, 2, 3
$endgroup$
– André 3000
Dec 2 '18 at 8:39
|
show 1 more comment
1
$begingroup$
Hey, and welcome to M.SE! To help you, it is helpful to know what you have tried, what your background is (e.g., are you comfortable with the notion of direct sum and cyclic module?), and what your process of thinking was when you approached this question.
$endgroup$
– Luke
Dec 1 '18 at 20:42
$begingroup$
@Luke, I'm learning modules for the first time. So that, when I think of a module I'm always with the generalization of a vector space. When I think of direct sum I always think on the linear combination of linearly independent generators of subspaces. But I don't know how can I relate this thinking with that question.
$endgroup$
– Mike Hawk
Dec 1 '18 at 22:29
$begingroup$
@MikeHawk you seem to have confused the title field for the tags field. I changed the title to be more useful. You should do something like that next time. Remember, the goal is to be informative and searchable. If you don’t, well, you’ll probably find out what the consequences are. Good luck.
$endgroup$
– rschwieb
Dec 2 '18 at 0:43
$begingroup$
Thank you @rschwieb
$endgroup$
– Mike Hawk
Dec 2 '18 at 0:56
$begingroup$
Similar to these: 1, 2, 3
$endgroup$
– André 3000
Dec 2 '18 at 8:39
1
1
$begingroup$
Hey, and welcome to M.SE! To help you, it is helpful to know what you have tried, what your background is (e.g., are you comfortable with the notion of direct sum and cyclic module?), and what your process of thinking was when you approached this question.
$endgroup$
– Luke
Dec 1 '18 at 20:42
$begingroup$
Hey, and welcome to M.SE! To help you, it is helpful to know what you have tried, what your background is (e.g., are you comfortable with the notion of direct sum and cyclic module?), and what your process of thinking was when you approached this question.
$endgroup$
– Luke
Dec 1 '18 at 20:42
$begingroup$
@Luke, I'm learning modules for the first time. So that, when I think of a module I'm always with the generalization of a vector space. When I think of direct sum I always think on the linear combination of linearly independent generators of subspaces. But I don't know how can I relate this thinking with that question.
$endgroup$
– Mike Hawk
Dec 1 '18 at 22:29
$begingroup$
@Luke, I'm learning modules for the first time. So that, when I think of a module I'm always with the generalization of a vector space. When I think of direct sum I always think on the linear combination of linearly independent generators of subspaces. But I don't know how can I relate this thinking with that question.
$endgroup$
– Mike Hawk
Dec 1 '18 at 22:29
$begingroup$
@MikeHawk you seem to have confused the title field for the tags field. I changed the title to be more useful. You should do something like that next time. Remember, the goal is to be informative and searchable. If you don’t, well, you’ll probably find out what the consequences are. Good luck.
$endgroup$
– rschwieb
Dec 2 '18 at 0:43
$begingroup$
@MikeHawk you seem to have confused the title field for the tags field. I changed the title to be more useful. You should do something like that next time. Remember, the goal is to be informative and searchable. If you don’t, well, you’ll probably find out what the consequences are. Good luck.
$endgroup$
– rschwieb
Dec 2 '18 at 0:43
$begingroup$
Thank you @rschwieb
$endgroup$
– Mike Hawk
Dec 2 '18 at 0:56
$begingroup$
Thank you @rschwieb
$endgroup$
– Mike Hawk
Dec 2 '18 at 0:56
$begingroup$
Similar to these: 1, 2, 3
$endgroup$
– André 3000
Dec 2 '18 at 8:39
$begingroup$
Similar to these: 1, 2, 3
$endgroup$
– André 3000
Dec 2 '18 at 8:39
|
show 1 more comment
0
active
oldest
votes
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',
autoActivateHeartbeat: false,
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
});
}
});
draft saved
draft discarded
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%2f3021785%2fhow-do-you-write-this-module-as-a-direct-sum-of-cyclic-modules%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
draft saved
draft discarded
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.
draft saved
draft discarded
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%2f3021785%2fhow-do-you-write-this-module-as-a-direct-sum-of-cyclic-modules%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
1
$begingroup$
Hey, and welcome to M.SE! To help you, it is helpful to know what you have tried, what your background is (e.g., are you comfortable with the notion of direct sum and cyclic module?), and what your process of thinking was when you approached this question.
$endgroup$
– Luke
Dec 1 '18 at 20:42
$begingroup$
@Luke, I'm learning modules for the first time. So that, when I think of a module I'm always with the generalization of a vector space. When I think of direct sum I always think on the linear combination of linearly independent generators of subspaces. But I don't know how can I relate this thinking with that question.
$endgroup$
– Mike Hawk
Dec 1 '18 at 22:29
$begingroup$
@MikeHawk you seem to have confused the title field for the tags field. I changed the title to be more useful. You should do something like that next time. Remember, the goal is to be informative and searchable. If you don’t, well, you’ll probably find out what the consequences are. Good luck.
$endgroup$
– rschwieb
Dec 2 '18 at 0:43
$begingroup$
Thank you @rschwieb
$endgroup$
– Mike Hawk
Dec 2 '18 at 0:56
$begingroup$
Similar to these: 1, 2, 3
$endgroup$
– André 3000
Dec 2 '18 at 8:39