In　this　paper,　the　inner　products　on　the　semimodules　over　a　commutative　semiring　are　investigated.　Some　characterizations　for　inner　products　and　for　orthogonal　sets　in　the　semimodules　are　given,　and　standard　orthogonal　basis　and　orthogonal　subsemimodule　are　discussed.　In　particular,　an　equivalent　description.　is　obtained　for　a　semifield　S　satisfying　the　property　that　every　standard　orthogonal　set　in　a　finitely　generated　semimodule　M　over　S　can　be　extended　to　a　standard　orthogonal　basis　for　M.　Also,　the　analogue　of　the　Kronecker-Capelli　theorem　is　proved　to　be　valid　for　a　matrix　equation　over　a　commutative　semiring.　Partial　results　obtained　in　this　paper　generalize　and　develop　corresponding　results　for　semilinear　spaces　of　n-dimensional　vectors　over　commutative　zerosum-free　semirings.　(C)　2014　Elsevier　Inc.　All　rights　reserved.