Second isomorphism theorem for modules, doubts with proofFurther explanation on proof that associated primes are precisely those belonging to primary modules in reduced decomposition of $0$.Error in proof that submodules of f.g. modules are f.g.Module isomorphism, simple modules, and quotientsRank-nullity theorem for modulesIsomorphism between modules and submodulesProblem with first isomorphism theorem for modulesPrimary decomposition of modules - uniqueness proofIsomorphism of modules and projectionProof of the Second Isomorphism Theorem for modulesTo prove the Second Isomorphism Theorem for modules from the opposite direction:
The baby cries all morning
Hide Select Output from T-SQL
Hostile work environment after whistle-blowing on coworker and our boss. What do I do?
How do I rename a LINUX host without needing to reboot for the rename to take effect?
Implement the Thanos sorting algorithm
Teaching indefinite integrals that require special-casing
Why are on-board computers allowed to change controls without notifying the pilots?
Why Were Madagascar and New Zealand Discovered So Late?
How can I use the arrow sign in my bash prompt?
Can criminal fraud exist without damages?
What is the opposite of 'gravitas'?
How does it work when somebody invests in my business?
Efficiently merge handle parallel feature branches in SFDX
Products and sum of cubes in Fibonacci
Is the destination of a commercial flight important for the pilot?
Can a monster with multiattack use this ability if they are missing a limb?
How was Earth single-handedly capable of creating 3 of the 4 gods of chaos?
Was the picture area of a CRT a parallelogram (instead of a true rectangle)?
Go Pregnant or Go Home
Bash method for viewing beginning and end of file
What to do with wrong results in talks?
Your magic is very sketchy
Will it be accepted, if there is no ''Main Character" stereotype?
Everything Bob says is false. How does he get people to trust him?
Second isomorphism theorem for modules, doubts with proof
Further explanation on proof that associated primes are precisely those belonging to primary modules in reduced decomposition of $0$.Error in proof that submodules of f.g. modules are f.g.Module isomorphism, simple modules, and quotientsRank-nullity theorem for modulesIsomorphism between modules and submodulesProblem with first isomorphism theorem for modulesPrimary decomposition of modules - uniqueness proofIsomorphism of modules and projectionProof of the Second Isomorphism Theorem for modulesTo prove the Second Isomorphism Theorem for modules from the opposite direction:
$begingroup$
Theorem:
Let $S$ and $T$ be submodules of $M$, and let $S+T=s+t,sin S,t in T$. Then $S+T$ and $S∩T$ are submodules of M, and:
$S/(S∩T)≃(S+T)/T$.
I saw the proof but don't we have to check also $S cap T$ is submodule of $S$ and $T$ is submodule of $S+T$? As I remember these terms have to be satisfied due to definition. Or maybe it is obvious for everyone?
abstract-algebra modules
asked Mar 17 at 13:35
ToidiToidi
254
$endgroup$
add a comment |
$begingroup$
Theorem:
Let $S$ and $T$ be submodules of $M$, and let $S+T=s+t,sin S,t in T$. Then $S+T$ and $S∩T$ are submodules of M, and:
$S/(S∩T)≃(S+T)/T$.
I saw the proof but don't we have to check also $S cap T$ is submodule of $S$ and $T$ is submodule of $S+T$? As I remember these terms have to be satisfied due to definition. Or maybe it is obvious for everyone?
abstract-algebra modules
asked Mar 17 at 13:35
ToidiToidi
254
$endgroup$
add a comment |
$begingroup$
Theorem:
Let $S$ and $T$ be submodules of $M$, and let $S+T=s+t,sin S,t in T$. Then $S+T$ and $S∩T$ are submodules of M, and:
$S/(S∩T)≃(S+T)/T$.
I saw the proof but don't we have to check also $S cap T$ is submodule of $S$ and $T$ is submodule of $S+T$? As I remember these terms have to be satisfied due to definition. Or maybe it is obvious for everyone?
abstract-algebra modules
asked Mar 17 at 13:35
ToidiToidi
254
$endgroup$
Theorem:
Let $S$ and $T$ be submodules of $M$, and let $S+T=s+t,sin S,t in T$. Then $S+T$ and $S∩T$ are submodules of M, and:
$S/(S∩T)≃(S+T)/T$.
I saw the proof but don't we have to check also $S cap T$ is submodule of $S$ and $T$ is submodule of $S+T$? As I remember these terms have to be satisfied due to definition. Or maybe it is obvious for everyone?
abstract-algebra modules
abstract-algebra modules
asked Mar 17 at 13:35
ToidiToidi
254
asked Mar 17 at 13:35
ToidiToidi
254
asked Mar 17 at 13:35
ToidiToidi
254
asked Mar 17 at 13:35
ToidiToidi
254
asked Mar 17 at 13:35
ToidiToidi
254
254
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It does need to be checked. It's rather easy though: They are subsets and they are modules, therefore they are submodules.
answered Mar 17 at 13:45
ArthurArthur
120k7120203
$endgroup$
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',
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
);
);
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%2f3151550%2fsecond-isomorphism-theorem-for-modules-doubts-with-proof%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
$begingroup$
It does need to be checked. It's rather easy though: They are subsets and they are modules, therefore they are submodules.
answered Mar 17 at 13:45
ArthurArthur
120k7120203
$endgroup$
add a comment |
$begingroup$
It does need to be checked. It's rather easy though: They are subsets and they are modules, therefore they are submodules.
answered Mar 17 at 13:45
ArthurArthur
120k7120203
$endgroup$
add a comment |
$begingroup$
It does need to be checked. It's rather easy though: They are subsets and they are modules, therefore they are submodules.
answered Mar 17 at 13:45
ArthurArthur
120k7120203
$endgroup$
It does need to be checked. It's rather easy though: They are subsets and they are modules, therefore they are submodules.
answered Mar 17 at 13:45
ArthurArthur
120k7120203
answered Mar 17 at 13:45
ArthurArthur
120k7120203
answered Mar 17 at 13:45
ArthurArthur
120k7120203
answered Mar 17 at 13:45
ArthurArthur
120k7120203
120k7120203
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.
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%2f3151550%2fsecond-isomorphism-theorem-for-modules-doubts-with-proof%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
