  In this work, we introduce the notion of $S$-1-absorbing primary submodule as
an extension of 1-absorbing primary submodule. Let $S$ be a multiplicatively
closed subset of a ring $R$ and $M$ be an $R$-module. A submodule $N$ of $M$
with $(N:_{R}M)\cap S=\emptyset$ is said to be $S$-1-absorbing primary if
whenever $abm\in N$ for some non-unit $a,b\in R$ and $m\in M$, then either
$sab\in(N:_{R}M)$ or $sm\in M$-$rad(N)$. We examine several properties of this
concept and provide some characterizations. In addition, $S$-1-absorbing
primary avoidance theorem and $S $-1-absorbing primary property for
idealization and amalgamation are presented.
