We study self-dual codes over the infinite family of rings R_k, which has been recently introduced to the literature.We prove that for each self-dual code over R_k, k \geq 2,  thereexist a corresponding binary self-dual code, a real unimodularlattice, a complex unimodular lattice, a quaternionic lattice and aninfinite family of self-dual codes. We prove the existence of TypeII codes of all lengths over R_k, for k\geq 3,  and we obtain someextremal binary self-dual codes including the extended binary Golaycode as the Gray images of self-dual codes over R_k for somesuitable k. The binary self-dual codes obtained from R_k all have automorphism groups whose orders are a multiple of 2^k.