Multiple commutator formulas

Let A be a quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. Let Ii, i = 0, . . . ,m, be two-sided ideals of A, GLn(A, Ii) be the principal congruence subgroup of level Ii in GLn(A) and En(A, Ii) be the relative elementary subgroup of level Ii. We prove the multiple commutator formula [En(A, I0), GLn(A, I1), GLn(A, I2), . . . ,GLn(A, Im)] = [En(A, I0), En(A, I1), En(A, I2), . . . ,En(A, Im)], 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.