# ra.rings and algebras – Are the trace relations among matrices generated by cyclic permutations?

Let $$X_1,dots,X_n$$ be non commutative variables such that $$operatorname{tr} f(X_1,dots,X_n) = 0$$ whenever the $$X_i$$ are specialized to square matrices in $$M_r(k)$$ for any $$r geq 1$$. Does this imply that $$f$$ is in the ideal generated by cyclic permutations: $$g_1dots g_k – g_2dots g_k g_1$$ for any polynomials $$g_i$$ in the $$X_i$$ and $$k geq 2$$?

(And if I have missed any obvious relations, is the statement true up to adding in those relations to the ideal?)