**Authors:** Manish Kant Dubey, Poonam Sarohe

### Abstract

In this paper, we define $2$-absorbing primary subsemimodules of a semimodule
$M$ over a commutative semiring $S$ with $1\neq 0$ 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.

Manish Kant Dubey

SAG, DRDO, Metcalf House, Delhi 110054, India.

E-mail:

Poonam Sarohe

Department of Mathematics, Lakshmibai College,

University of Delhi, Delhi 110052, India.

E-mail:

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