Some algorithmic questions of associative algebras. (Q1397576)

Let \(A=\langle X\mid f_1,\dots,f_m\rangle\) be a finitely presented associative algebra over a field of characteristic 0, where \(\{f_1,\dots,f_m\}\) is a Gröbner (Gröbner-Shirshov) basis. Then \(A\) is called a canonically finitely presented algebra. Let \(B=\langle g_1,\dots,g_N\rangle\) be a subalgebra of \(A\) generated by a SAGBI-basis (Subalgebra Analogue of Gröbner Bases for Ideals) \(\{g_1,\dots,g_N\}\) [\textit{D. Kapur} and \textit{K. Madlener}, Computers and mathematics, Proc. Conf., Cambridge/Mass. 1989, 1-11 (1989; Zbl 0692.13001), \textit{L. Robbiano} and \textit{M. Sweedler}, Lect. Notes Math. 1430, 61-87 (1990; Zbl 0725.13013)]. The following property of \(B\) is algorithmically recognisable: ``\(B\) is a free subalgebra with the free generators \(g_1,\dots,g_N\)''.