$phi$-primary subsemimodules

Let $r$ be a commutative semiring with identity and $m$ be a unitary $r$-semimodule. ##let $phi:mathcal{s}(m)rightarrow mathcal{s}(m)cup{emptyset}$ be a function, where $mathcal{s}(m)$ is the set of all subsemimodules of $m$. ##a proper subsemimodule $n$ of $m$ is called $phi$-primary subsemimodule, if whenever $rin r$ and $xin m$ with $rxin n-phi(n)$, implies that $rin sqrt{(n:_r m)}$ or $xin n$. so if we take $phi(n)=emptyset$ (resp., $phi(n)={0}$), a $phi$-primary subsemimodule is primary (resp., weakly primary). in this paper, we study the concept of $phi$-primary subsemimodule which is a generalization of $phi$-prime subsemimodule in a commutative semiring