Can I Find A Map from a Module M to the kernel of a map p from M to M?
I have a module homomorphism $p:Mrightarrow M$. I would like to find another module homomorphism $phi:Mrightarrow ker(p)$. Finding such a thing seems to be very challenging however. Is this possible? Note also that $p^2=p$.
modules
edited Nov 25 at 20:16
Ashwin Trisal
1,1791516
asked Nov 25 at 20:09
V. P. Sworski
11
add a comment |
I have a module homomorphism $p:Mrightarrow M$. I would like to find another module homomorphism $phi:Mrightarrow ker(p)$. Finding such a thing seems to be very challenging however. Is this possible? Note also that $p^2=p$.
modules
edited Nov 25 at 20:16
Ashwin Trisal
1,1791516
asked Nov 25 at 20:09
V. P. Sworski
11
add a comment |
I have a module homomorphism $p:Mrightarrow M$. I would like to find another module homomorphism $phi:Mrightarrow ker(p)$. Finding such a thing seems to be very challenging however. Is this possible? Note also that $p^2=p$.
modules
edited Nov 25 at 20:16
Ashwin Trisal
1,1791516
asked Nov 25 at 20:09
V. P. Sworski
11
I have a module homomorphism $p:Mrightarrow M$. I would like to find another module homomorphism $phi:Mrightarrow ker(p)$. Finding such a thing seems to be very challenging however. Is this possible? Note also that $p^2=p$.
modules
modules
edited Nov 25 at 20:16
Ashwin Trisal
1,1791516
asked Nov 25 at 20:09
V. P. Sworski
11
edited Nov 25 at 20:16
Ashwin Trisal
1,1791516
asked Nov 25 at 20:09
V. P. Sworski
11
edited Nov 25 at 20:16
Ashwin Trisal
1,1791516
edited Nov 25 at 20:16
Ashwin Trisal
1,1791516
edited Nov 25 at 20:16
Ashwin Trisal
1,1791516
1,1791516
asked Nov 25 at 20:09
V. P. Sworski
11
asked Nov 25 at 20:09
V. P. Sworski
11
asked Nov 25 at 20:09
V. P. Sworski
11
11
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
Let $phi=1_M-p$. Then $pcirc phi=p(1_M-p)=p-p^2=p-p=0$, so $phi:Mtoker p$. But there are many such maps $phi$; taking $phi$ to be the $0$ map would also define a map from $M$ to $ker p$.
answered Nov 25 at 20:15
Ashwin Trisal
1,1791516
Ah, I see what I'm missing. I need the map to be surjective as well.
– V. P. Sworski
Nov 25 at 20:25
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%2f3013324%2fcan-i-find-a-map-from-a-module-m-to-the-kernel-of-a-map-p-from-m-to-m%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
Let $phi=1_M-p$. Then $pcirc phi=p(1_M-p)=p-p^2=p-p=0$, so $phi:Mtoker p$. But there are many such maps $phi$; taking $phi$ to be the $0$ map would also define a map from $M$ to $ker p$.
answered Nov 25 at 20:15
Ashwin Trisal
1,1791516
Ah, I see what I'm missing. I need the map to be surjective as well.
– V. P. Sworski
Nov 25 at 20:25
add a comment |
Let $phi=1_M-p$. Then $pcirc phi=p(1_M-p)=p-p^2=p-p=0$, so $phi:Mtoker p$. But there are many such maps $phi$; taking $phi$ to be the $0$ map would also define a map from $M$ to $ker p$.
answered Nov 25 at 20:15
Ashwin Trisal
1,1791516
Ah, I see what I'm missing. I need the map to be surjective as well.
– V. P. Sworski
Nov 25 at 20:25
add a comment |
Let $phi=1_M-p$. Then $pcirc phi=p(1_M-p)=p-p^2=p-p=0$, so $phi:Mtoker p$. But there are many such maps $phi$; taking $phi$ to be the $0$ map would also define a map from $M$ to $ker p$.
answered Nov 25 at 20:15
Ashwin Trisal
1,1791516
Let $phi=1_M-p$. Then $pcirc phi=p(1_M-p)=p-p^2=p-p=0$, so $phi:Mtoker p$. But there are many such maps $phi$; taking $phi$ to be the $0$ map would also define a map from $M$ to $ker p$.
answered Nov 25 at 20:15
Ashwin Trisal
1,1791516
answered Nov 25 at 20:15
Ashwin Trisal
1,1791516
answered Nov 25 at 20:15
Ashwin Trisal
1,1791516
answered Nov 25 at 20:15
Ashwin Trisal
1,1791516
1,1791516
Ah, I see what I'm missing. I need the map to be surjective as well.
– V. P. Sworski
Nov 25 at 20:25
add a comment |
Ah, I see what I'm missing. I need the map to be surjective as well.
– V. P. Sworski
Nov 25 at 20:25
Ah, I see what I'm missing. I need the map to be surjective as well.
– V. P. Sworski
Nov 25 at 20:25
Ah, I see what I'm missing. I need the map to be surjective as well.
– V. P. Sworski
Nov 25 at 20:25
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%2f3013324%2fcan-i-find-a-map-from-a-module-m-to-the-kernel-of-a-map-p-from-m-to-m%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
