A class of array codes over rings of integers modulo-q with good Euclidean distance properties is introduced. Depending on the design, these codes can have linear or nonlinear properties. An extension of a simple algorithm to design a low-complexity trellis diagram for array codes over GF(2) introduced recently is developed for array codes over rings. These codes over rings are compared to the corresponding codes over GF(2), where particular attention is given to the coding gain, spectral efficiency, codebook size and trellis complexity. It is shown that array codes over Z4 and Z8 provide a twofold and threefold increase, respectively, in spectral efficiency as well as a higher coding gain over uncoded transmission and a much larger codebook than that obtained with the same array codes over GF(2) for similar code parameters.