Relative Centralizers of Relative Subgroups

Let R be an associative ring with 1 and G = GL(n, R) the general linear group of degree n ≥ 3 over R. A goal of the paper is to calculate the relative centralizers of the relative elementary subgroups or the principal congruence subgroups, corresponding to an ideal A ⊴ R modulo the relative elementary subgroups or the principal congruence subgroups, corresponding to another ideal B ⊴ R. Modulo congruence subgroups, the results are essentially easy exercises in linear algebra. But modulo the elementary subgroups, they turned out to be quite tricky, and definitive answers are obtained only over commutative rings or, in some cases, only over Dedekind rings/Dedekind rings of arithmetic type.