In this paper, we introduce a partitioning subsemimodule of a semimodule over a semiring
which is useful to develop the quotient structure of semimodule. Indeed we prove : 1) The quotient semimodule
$ M/N_{(Q)}$ is essentially independent of
choice of $Q$. 2) If $ f :M ightarrow M^{'}$ is a maximal
$R$-semimodule homomorphism, then $M/kerf_{(Q)} cong M^{'}$. 3)
Every partitioning subsemimodule is subtractive. 4) Let $N$ be a
$Q$-subsemimodule of an $R$-semimodule $M$. Then $A$ is a
subtractive subsemimodule of $M$ with $N subseteq A$ if and only
if $A/N_{( Q cap A)}$ = ${q + N : q in Q cap A}$ is a
subtractive subsemimodule of $M/N_{(Q)}$.

Keywords: semimodule, subtractive subsemimodule, partitioning subsemi-
module, quotient semimodule, maximal homomorphism, isomorphism