Ideals generated and their inclusions
$begingroup$
Let $R=2mathbb{Z}$, then $R$ is a ring without identity, and we consider the ideal generated by $4$, that is $$(4)={4(2j)+4overline{k};|;j,overline{k}inmathbb{Z}}.$$
I must prove that $(4)=4mathbb{Z}$. Now, I proved that $(4)subseteq4mathbb{Z}$, in fact, let $xin(4)$, then $x=4(2j)+4overline{k}=4(2j+overline{k})in 4mathbb{Z}$.
Can I prove that $4mathbb{Z}subseteq (4)$? Thanks!
abstract-algebra proof-verification proof-writing proof-explanation
asked Dec 2 '18 at 15:58
Jack J.Jack J.
4672419
$endgroup$
add a comment |
$begingroup$
Let $R=2mathbb{Z}$, then $R$ is a ring without identity, and we consider the ideal generated by $4$, that is $$(4)={4(2j)+4overline{k};|;j,overline{k}inmathbb{Z}}.$$
I must prove that $(4)=4mathbb{Z}$. Now, I proved that $(4)subseteq4mathbb{Z}$, in fact, let $xin(4)$, then $x=4(2j)+4overline{k}=4(2j+overline{k})in 4mathbb{Z}$.
Can I prove that $4mathbb{Z}subseteq (4)$? Thanks!
abstract-algebra proof-verification proof-writing proof-explanation
asked Dec 2 '18 at 15:58
Jack J.Jack J.
4672419
$endgroup$
add a comment |
$begingroup$
Let $R=2mathbb{Z}$, then $R$ is a ring without identity, and we consider the ideal generated by $4$, that is $$(4)={4(2j)+4overline{k};|;j,overline{k}inmathbb{Z}}.$$
I must prove that $(4)=4mathbb{Z}$. Now, I proved that $(4)subseteq4mathbb{Z}$, in fact, let $xin(4)$, then $x=4(2j)+4overline{k}=4(2j+overline{k})in 4mathbb{Z}$.
Can I prove that $4mathbb{Z}subseteq (4)$? Thanks!
abstract-algebra proof-verification proof-writing proof-explanation
asked Dec 2 '18 at 15:58
Jack J.Jack J.
4672419
$endgroup$
Let $R=2mathbb{Z}$, then $R$ is a ring without identity, and we consider the ideal generated by $4$, that is $$(4)={4(2j)+4overline{k};|;j,overline{k}inmathbb{Z}}.$$
I must prove that $(4)=4mathbb{Z}$. Now, I proved that $(4)subseteq4mathbb{Z}$, in fact, let $xin(4)$, then $x=4(2j)+4overline{k}=4(2j+overline{k})in 4mathbb{Z}$.
Can I prove that $4mathbb{Z}subseteq (4)$? Thanks!
abstract-algebra proof-verification proof-writing proof-explanation
abstract-algebra proof-verification proof-writing proof-explanation
asked Dec 2 '18 at 15:58
Jack J.Jack J.
4672419
asked Dec 2 '18 at 15:58
Jack J.Jack J.
4672419
asked Dec 2 '18 at 15:58
Jack J.Jack J.
4672419
asked Dec 2 '18 at 15:58
Jack J.Jack J.
4672419
asked Dec 2 '18 at 15:58
Jack J.Jack J.
4672419
4672419
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let $x$ be a multiple of $4$, so $x=4k$, with integer $k$. Then
$$x=4k=4(2cdot0)+4kin(4)$$
answered Dec 2 '18 at 16:13
egregegreg
182k1485204
$endgroup$
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%2f3022802%2fideals-generated-and-their-inclusions%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
$begingroup$
Let $x$ be a multiple of $4$, so $x=4k$, with integer $k$. Then
$$x=4k=4(2cdot0)+4kin(4)$$
answered Dec 2 '18 at 16:13
egregegreg
182k1485204
$endgroup$
add a comment |
$begingroup$
Let $x$ be a multiple of $4$, so $x=4k$, with integer $k$. Then
$$x=4k=4(2cdot0)+4kin(4)$$
answered Dec 2 '18 at 16:13
egregegreg
182k1485204
$endgroup$
add a comment |
$begingroup$
Let $x$ be a multiple of $4$, so $x=4k$, with integer $k$. Then
$$x=4k=4(2cdot0)+4kin(4)$$
answered Dec 2 '18 at 16:13
egregegreg
182k1485204
$endgroup$
Let $x$ be a multiple of $4$, so $x=4k$, with integer $k$. Then
$$x=4k=4(2cdot0)+4kin(4)$$
answered Dec 2 '18 at 16:13
egregegreg
182k1485204
answered Dec 2 '18 at 16:13
egregegreg
182k1485204
answered Dec 2 '18 at 16:13
egregegreg
182k1485204
answered Dec 2 '18 at 16:13
egregegreg
182k1485204
182k1485204
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%2f3022802%2fideals-generated-and-their-inclusions%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
