Proving the $(1)$ is a prime element in the set T
$begingroup$
I am wondering if there are any logical leaps and anyway for the proof to be concise.
Background information:
$T=left{left[begin{array}{cc}
a & b \
c & d
end{array}right]:, a,b,c,din mathbb{Z},, a=b=c=d right }$
and the notation for an element in $T$:
$(x)=left[begin{array}{cc}
x & x \
x & x
end{array}right], forall xin mathbb{Z}$
$(x)(y)=(2xy)$
The definition of being composite and prime in $T$:
An element $(x)$ is $textbf{composite}$ in $T$ if and only if $(x)=(y)(z)$ where $(y),(z) in T$. If $(x)$ is not composite, it is said to be $textbf{prime}$.
Proof:
Suppose $(1)$ is a composite. Then by the above Definition: $(1)=(2xy)=(x)(y)$ where $x,yin mathbb{Z}$ and $yneq0$. Thus, $1=2xy$. $x=frac{1}{2y}$. Then, $xin mathbb{Q}$ because $y in mathbb{Z}$. But $x$ is defined in $mathbb{Z}$. Therefore, $x notin mathbb{Z} implies (x)notin T$. Therefore, $(1)$ is a prime element in $T$.
Citation: Jacobson, Bernard. “Matrix Number Theory: An Example of Nonunique Factorization.” The American Mathematical Monthly, vol. 72, no. 4, 1965, pp. 399–402. JSTOR, www.jstor.org/stable/2313504.
Thanks,
Cba
matrices
asked Dec 9 '18 at 23:30
cbacba
13
$endgroup$
add a comment |
$begingroup$
I am wondering if there are any logical leaps and anyway for the proof to be concise.
Background information:
$T=left{left[begin{array}{cc}
a & b \
c & d
end{array}right]:, a,b,c,din mathbb{Z},, a=b=c=d right }$
and the notation for an element in $T$:
$(x)=left[begin{array}{cc}
x & x \
x & x
end{array}right], forall xin mathbb{Z}$
$(x)(y)=(2xy)$
The definition of being composite and prime in $T$:
An element $(x)$ is $textbf{composite}$ in $T$ if and only if $(x)=(y)(z)$ where $(y),(z) in T$. If $(x)$ is not composite, it is said to be $textbf{prime}$.
Proof:
Suppose $(1)$ is a composite. Then by the above Definition: $(1)=(2xy)=(x)(y)$ where $x,yin mathbb{Z}$ and $yneq0$. Thus, $1=2xy$. $x=frac{1}{2y}$. Then, $xin mathbb{Q}$ because $y in mathbb{Z}$. But $x$ is defined in $mathbb{Z}$. Therefore, $x notin mathbb{Z} implies (x)notin T$. Therefore, $(1)$ is a prime element in $T$.
Citation: Jacobson, Bernard. “Matrix Number Theory: An Example of Nonunique Factorization.” The American Mathematical Monthly, vol. 72, no. 4, 1965, pp. 399–402. JSTOR, www.jstor.org/stable/2313504.
Thanks,
Cba
matrices
asked Dec 9 '18 at 23:30
cbacba
13
$endgroup$
add a comment |
$begingroup$
I am wondering if there are any logical leaps and anyway for the proof to be concise.
Background information:
$T=left{left[begin{array}{cc}
a & b \
c & d
end{array}right]:, a,b,c,din mathbb{Z},, a=b=c=d right }$
and the notation for an element in $T$:
$(x)=left[begin{array}{cc}
x & x \
x & x
end{array}right], forall xin mathbb{Z}$
$(x)(y)=(2xy)$
The definition of being composite and prime in $T$:
An element $(x)$ is $textbf{composite}$ in $T$ if and only if $(x)=(y)(z)$ where $(y),(z) in T$. If $(x)$ is not composite, it is said to be $textbf{prime}$.
Proof:
Suppose $(1)$ is a composite. Then by the above Definition: $(1)=(2xy)=(x)(y)$ where $x,yin mathbb{Z}$ and $yneq0$. Thus, $1=2xy$. $x=frac{1}{2y}$. Then, $xin mathbb{Q}$ because $y in mathbb{Z}$. But $x$ is defined in $mathbb{Z}$. Therefore, $x notin mathbb{Z} implies (x)notin T$. Therefore, $(1)$ is a prime element in $T$.
Citation: Jacobson, Bernard. “Matrix Number Theory: An Example of Nonunique Factorization.” The American Mathematical Monthly, vol. 72, no. 4, 1965, pp. 399–402. JSTOR, www.jstor.org/stable/2313504.
Thanks,
Cba
matrices
asked Dec 9 '18 at 23:30
cbacba
13
$endgroup$
I am wondering if there are any logical leaps and anyway for the proof to be concise.
Background information:
$T=left{left[begin{array}{cc}
a & b \
c & d
end{array}right]:, a,b,c,din mathbb{Z},, a=b=c=d right }$
and the notation for an element in $T$:
$(x)=left[begin{array}{cc}
x & x \
x & x
end{array}right], forall xin mathbb{Z}$
$(x)(y)=(2xy)$
The definition of being composite and prime in $T$:
An element $(x)$ is $textbf{composite}$ in $T$ if and only if $(x)=(y)(z)$ where $(y),(z) in T$. If $(x)$ is not composite, it is said to be $textbf{prime}$.
Proof:
Suppose $(1)$ is a composite. Then by the above Definition: $(1)=(2xy)=(x)(y)$ where $x,yin mathbb{Z}$ and $yneq0$. Thus, $1=2xy$. $x=frac{1}{2y}$. Then, $xin mathbb{Q}$ because $y in mathbb{Z}$. But $x$ is defined in $mathbb{Z}$. Therefore, $x notin mathbb{Z} implies (x)notin T$. Therefore, $(1)$ is a prime element in $T$.
Citation: Jacobson, Bernard. “Matrix Number Theory: An Example of Nonunique Factorization.” The American Mathematical Monthly, vol. 72, no. 4, 1965, pp. 399–402. JSTOR, www.jstor.org/stable/2313504.
Thanks,
Cba
matrices
matrices
asked Dec 9 '18 at 23:30
cbacba
13
asked Dec 9 '18 at 23:30
cbacba
13
edited Dec 9 '18 at 23:42
cba
asked Dec 9 '18 at 23:30
cbacba
13
asked Dec 9 '18 at 23:30
cbacba
13
asked Dec 9 '18 at 23:30
cbacba
13
13
add a comment |
add a 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
});
}
});
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%2f3033184%2fproving-the-1-is-a-prime-element-in-the-set-t%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
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%2f3033184%2fproving-the-1-is-a-prime-element-in-the-set-t%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
