Let S be an inverse semiring, and U be an ideal of S. In this paper, we introduce the concept of U-S Jordan homomorphism of inverse semirings, and extend the result of Herstein on Jordan homomorphisms in inverse semirings.
Let R be a commutative ring with unity and an R-submodule N is called semimaximal if and only if
the sufficient conditions of F-submodules to be semimaximal .Also the concepts of (simple , semisimple) F- submodules and quotient F- modules are introduced and given some properties .