ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 27 Feb 2019 03:50:26 -0600Quotient of non-commutative ringhttp://ask.sagemath.org/question/45480/quotient-of-non-commutative-ring/ I'm trying to make a non-commutative ring, where X^2 = -1 ; XY = -YX and Y^2 = 0, something like this ;
R[X,Y] / (X^2 + 1 , XY + YX , Y^2)
To get the non-commutative part, I did :
A.<x,y> = FreeAlgebra(QQ,2)
P.<x,y> = A.g_algebra(relations={y*x:-x*y})
What is the next command to have the quotient by (X^2+1 , Y^2) ? Or maybe an other way to build the entire ring, if there exists a simpler method :) thanks !
Sat, 16 Feb 2019 14:54:16 -0600http://ask.sagemath.org/question/45480/quotient-of-non-commutative-ring/Answer by rburing for <p>I'm trying to make a non-commutative ring, where X^2 = -1 ; XY = -YX and Y^2 = 0, something like this ; </p>
<pre><code> R[X,Y] / (X^2 + 1 , XY + YX , Y^2)
</code></pre>
<p>To get the non-commutative part, I did :</p>
<p>A.<x,y> = FreeAlgebra(QQ,2)</p>
<p>P.<x,y> = A.g_algebra(relations={y<em>x:-x</em>y})</p>
<p>What is the next command to have the quotient by (X^2+1 , Y^2) ? Or maybe an other way to build the entire ring, if there exists a simpler method :) thanks !</p>
http://ask.sagemath.org/question/45480/quotient-of-non-commutative-ring/?answer=45488#post-id-45488You want to quotient by the two-sided ideal generated by `x^2 + 1` and `y^2`:
R.<x,y> = P.quo(P*[x^2 + 1, y^2]*P)
Note this re-defines `x` and `y` once again, as you also did in your code.
Then you get:
sage: x^2
-1
sage: y^2
0
sage: y*x
-x*yMon, 18 Feb 2019 03:03:52 -0600http://ask.sagemath.org/question/45480/quotient-of-non-commutative-ring/?answer=45488#post-id-45488Comment by franslag for <p>You want to quotient by the two-sided ideal generated by <code>x^2 + 1</code> and <code>y^2</code>:</p>
<pre><code>R.<x,y> = P.quo(P*[x^2 + 1, y^2]*P)
</code></pre>
<p>Note this re-defines <code>x</code> and <code>y</code> once again, as you also did in your code.</p>
<p>Then you get:</p>
<pre><code>sage: x^2
-1
sage: y^2
0
sage: y*x
-x*y
</code></pre>
http://ask.sagemath.org/question/45480/quotient-of-non-commutative-ring/?comment=45558#post-id-45558This was helpful, thanks !!Wed, 27 Feb 2019 03:50:26 -0600http://ask.sagemath.org/question/45480/quotient-of-non-commutative-ring/?comment=45558#post-id-45558