For a semimodule over any semiring, the maximal and prime [Formula: see text]-subsemimodules are characterized with the help of the quotient structure. The main results are as follows. A semimodule is [Formula: see text]-congruence-simple iff it is [Formula: see text]-subsemimodule-simple. A nonzero semimodule is [Formula: see text]-simple iff it satisfies condition [Formula: see text]. A proper [Formula: see text]-subsemimodule of a semimodule is a maximal [Formula: see text]-subsemimodule iff the quotient semimodule is [Formula: see text]-simple. A proper [Formula: see text]-subsemimodule is prime in a semimodule iff the quotient semimodule is prime. A nonzero [Formula: see text]-simple semimodule over a commutative semiring is prime. Lemma 6.2 in this paper corrects the statement and proof of Lemma 3.16 in [Yeşilot Hacet. J. Math. Stat. 39 (2010) 305–312].
