weakly prime ternary subsemimodules of ternary semimodules

in this paper we introduce the concept of weakly prime ternary subsemimodules of a ternary semimodule over a ternary semiring and obtain some characterizations of weakly prime ternary subsemimodules. we prove that if $n$ is a weakly prime subtractive ternary subsemimodule of a ternary $r$-semimodule $m$, then either $n$ is a prime ternary subsemimodule or $(n : m)(n : m)n = 0$. if $n$ is a $q$-ternary subsemimodule of  a ternary $r$-semimodule $m$, then a relation between weakly prime ternary subsemimodules of $m$ containing $n$ and weakly prime ternary subsemimodules of the quotient ternary $r$-semimodule $m/n_{(q)}$ is obtained.