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.Fri, 18 Oct 2013 08:19:35 -0500Ideals of non-commutative polynomialshttp://ask.sagemath.org/question/9748/ideals-of-non-commutative-polynomials/Basically I have the same question as [here](http://ask.sagemath.org/question/1267/find-specific-linear-combination-in-multivariate), but in the non-commutative case: Given non-commutative polynomials $f_1,\dotsc,f_s \in \mathbb{Q}\langle x_1,\dotsc,x_n \rangle$, how can I test (with sage, or any other program which can do this) that some $g \in \mathbb{Q}\langle x_1,\dotsc,x_n \rangle$ satisfies $g \in \langle f_1,\dotsc,f_s \rangle$ (two-sided ideal), and find an explicit linear combination $g = \sum_i a_i f_i b_i$ which demonstrates this?
In trac ticket [#11068](http://trac.sagemath.org/sage_trac/ticket/11068) non-commutative quotient rings were implemented. However, according to the reference manual on [quotient rings](http://www.sagemath.org/doc/reference/sage/rings/quotient_ring.html), this assumes that one defines a reduce method by hand. But in my example , it is not clear how to do this.Sun, 27 Jan 2013 13:23:38 -0600http://ask.sagemath.org/question/9748/ideals-of-non-commutative-polynomials/Answer by Starx for <p>Basically I have the same question as <a href="http://ask.sagemath.org/question/1267/find-specific-linear-combination-in-multivariate">here</a>, but in the non-commutative case: Given non-commutative polynomials $f_1,\dotsc,f_s \in \mathbb{Q}\langle x_1,\dotsc,x_n \rangle$, how can I test (with sage, or any other program which can do this) that some $g \in \mathbb{Q}\langle x_1,\dotsc,x_n \rangle$ satisfies $g \in \langle f_1,\dotsc,f_s \rangle$ (two-sided ideal), and find an explicit linear combination $g = \sum_i a_i f_i b_i$ which demonstrates this?</p>
<p>In trac ticket <a href="http://trac.sagemath.org/sage_trac/ticket/11068">#11068</a> non-commutative quotient rings were implemented. However, according to the reference manual on <a href="http://www.sagemath.org/doc/reference/sage/rings/quotient_ring.html">quotient rings</a>, this assumes that one defines a reduce method by hand. But in my example , it is not clear how to do this.</p>
http://ask.sagemath.org/question/9748/ideals-of-non-commutative-polynomials/?answer=15558#post-id-15558What you're asking for doesn't exist in complete generality because this would imply a solution to the word problem. The GAP package GBNP implements Grobner bases for non-commutative polynomial rings. The algorithm need not terminate, but when it does it solves the problem you're asking about.Fri, 18 Oct 2013 08:19:35 -0500http://ask.sagemath.org/question/9748/ideals-of-non-commutative-polynomials/?answer=15558#post-id-15558