On $2$-absorbing primary subsemimodules over commutative semirings

In this paper, we define $2$-absorbing primary subsemimodules of a semimodule $M$ over a commutative semiring $S$ with $1\neq0$ which is a generalization of primary subsemimodules of semimodules. A proper subsemimodule $N$ of a semimodule $M$ is said to be a $2$-absorbing primary subsemimodule of $M$ if $abm\in N$ implies $ab\in \sqrt{(N:M)}$ or $am\in N$ or $bm\in N$ for some $a,b\in S$ and $m\in M$. It is proved that if $K$ is a subtractive subsemimodule of $M$ and $\sqrt{(K:M)}$ is a subtractive ideal of $S$, then $K$ is a $2$-absorbing primary subsemimodule of $M$ if and only if whenever $IJN\subseteq K$ for some ideals $I, J$ of $S$ and a subsemimodule $N$ of $M$, then $IJ\subseteq\sqrt{(K:M)}$ or $IN\subseteq K$ or $JN\subseteq K$. In this paper, we prove a number of results concerning $2$-absorbing primary subsemimodules.