# Ideals of non-commutative polynomials

Basically I have the same question as here, 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 non-commutative quotient rings were implemented. However, according to the reference manual on quotient rings, this assumes that one defines a reduce method by hand. But in my example , it is not clear how to do this.

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What 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.

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