Broadcasting K independent messages to multiple users where each user has a subset of the K messages as side information is studied. This problem can be regarded as a natural generalization of the well-known index coding problem to the physical-layer additive white Gaussian noise channel due to the analogy between these two problems. Recently, Natarajan, Hong, and Viterbo proposed a novel broadcasting strategy called lattice index coding which uses lattices constructed over principal ideal domains (PIDs) as a transmission scheme and showed that such a scheme provides uniform side information gains. In this paper, we generalize this strategy to rings of algebraic integers of number fields which may not be PIDs and show upper and lower bounds on the achievable side information gains. This generalization substantially enlarges the design space and includes some interesting examples in which all the messages are from the same field.