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Finite and infinite time ruin probabilities in a discrete time risk process with a Markov chain interest model are studied. Recursive and integral equations for the ruin probabilities are given. When interest rates are non-negative, generalized Lundberg inequalities for the infinite time ruin probability are derived by inductive and martingale approaches. When interest rates can be negative and loss distributions have regularly varying tails, asymptotic formulas for the finite time ruin probability are given by an inductive approach on the recursive equations.