MacWilliams identities of linear codes over non-principal ideal ring F_{_p}+vF_{_p}

We study the structure of the linear codes over ring F_{_p}+vF_{_p}, and prove that the Gray images of the dual codes are also dual. We define counting formulas of Lee weight, Hamming weight and generalized systematic weight distributions of linear codes over ring F_{_p}+vF_{_p}. By the relationship of linear codes and their dual codes over F_{_p} and the proposition of the Gray map, we give the MacWilliams identities between the linear codes and their dual codes. According to the identities, we can get the weight distributions of the dual codes directly without obtaining the dual codes of linear codes over ring F_{_p}+vF_{_p}, which can instruct us to have a thorough understanding for the inner structure of codes over ring F_{_p}+vF_{_p} and their Gray images.