On classical 2-prime subsemimodules

Abstract

In this paper, we introduce the notion of classical [Formula: see text]-prime subsemimodules of a semimodule [Formula: see text] over a commutative semiring [Formula: see text] with identity, which is a generalization of classical prime subsemimodules. A proper subsemimodule [Formula: see text] of [Formula: see text] having the property that for each [Formula: see text] and each subsemimodule [Formula: see text] of [Formula: see text], the inclusion [Formula: see text] implies [Formula: see text] or [Formula: see text] is called classical [Formula: see text]-prime. We give some results concerning classical [Formula: see text]-prime subsemimodules. Also, the classical [Formula: see text]-prime avoidance theorem for subsemimodules is proved. Finally, the notion of valuation semimodules is introduced and some results on classical [Formula: see text]-prime subsemimodules in valuation semimodules are given.