Multiple commutator formulas

Let A be a quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. Let I _i, i = 0,...,m, be two-sided ideals of A, GL_n(A, I _i) be the principal congruence subgroup of level I _i in GL_n(A) and E _n(A, I _i) be the relative elementary subgroup of level I _i. We prove the multiple commutator formula [E_n(A, I₀), GL_n(A, I₁), GL_n(A, I₂), ... , GL_n(A, I_m)] = [E_n(A, I₀), E_n(A, I₁), E_n(A, I₂), ... , E_n(A, I_m)], which is a broad generalization of the standard commutator formulas. This result contains all the published results on commutator formulas over commutative rings and answers a problem posed by A. Stepanov and N. Vavilov.