Geodesic
Relative centralisers of relative subgroups
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 35, Tome 492 (2020), pp. 10-24

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Let $R$ be an associative ring with $1$, $G=\mathrm{GL}(n, R)$ 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/Dedekind rings of arithmetic type. Bibliography: 43 titles. 
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     author = {N. A. Vavilov and Z. Zhang},
     title = {Relative centralisers of relative subgroups},
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N. A. Vavilov; Z. Zhang. Relative centralisers of relative subgroups. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 35, Tome 492 (2020), pp. 10-24. http://geodesic.mathdoc.fr/item/ZNSL_2020_492_a1/