  Let $R$ be an associative ring with 1, $G=GL(n, R)$ be the general linear
group of degree $n\ge 3$ over $R$. In this paper we calculate the relative
centralisers of the relative elementary subgroups or the principal congruence
subgroups, corresponding to an ideal $A\unlhd R$ modulo the relative elementary
subgroups or the principal congruence subgroups, corresponding to another ideal
$B\unlhd R$. 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 we could get definitive answers only over
commutative rings, or, in some cases, only over Dedekind rings. We discuss also
some further related problems, such as the interrelations of various birelative
commutator subgroups, etc., and state several unsolved questions.
