A Menon-type identity in residually finite Dedekind domains

Let φ be the Euler's totient function and σ k ( n ) = ∑ d | n d k , that is, the sum of the kth powers of the divisors of n. B. Sury showed that ∑ t 1 ∈ U ( Z n ) , t 2 , … , t r ∈ Z n gcd ( t 1 − 1 , t 2 , … , t r , n ) = φ ( n ) σ r − 1 ( n ) , where U ( Z n ) is the group of units in the ring of residual classes modulo n. Here, this identity is extended to residually finite Dedekind domains.