Primary decomposition of A-ideals in MV-semimodules

In [16], by using an MV-semiring and an MV-algebra, we introduced the new definition  of MV-semimodule and studied some of their basic properties. In this paper, we study  and present definitions of primary ideals of MV-semirings, decomposition of ideals in  MV-semirings, primary A-ideals of MV -semimodules, and decomposition of A-ideals in MV-semimodules. Then we present some conditions that an A-ideal can have a reduced primary decomposition.