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In this paper, we introduce the notion of classical 2-prime subsemimodules of a semimodule M over a commutative semiring S with identity, which is a generalization of classical prime subsemimodules. A proper subsemimodule N of M having the property that for each [Formula: see text] and each subsemimodule K of M, the inclusion [Formula: see text] implies [Formula: see text] or [Formula: see text] is called classical 2-prime. We give some results concerning classical 2-prime subsemimodules. Also, the classical 2-prime avoidance theorem for subsemimodules is proved. Finally, the notion of valuation semimodules is introduced and some results on classical 2-prime subsemimodules in valuation semimodules are given.

Let [Formula: see text] be a commutative ring with identity. In this paper, we introduce the concept of [Formula: see text]-absorbing prime ideals which is a generalization of prime ideals. A proper ideal [Formula: see text] of [Formula: see text] is called [Formula: see text]-absorbing prime if for all nonunit elements [Formula: see text] such that [Formula: see text], then either [Formula: see text] or [Formula: see text]. Some properties of [Formula: see text]-absorbing prime are studied. For instance, it is shown that if [Formula: see text] admits a [Formula: see text]-absorbing prime ideal that is not a prime ideal, then [Formula: see text] is a quasi–local ring. Among other things, it is proved that a proper ideal [Formula: see text] of [Formula: see text] is [Formula: see text]-absorbing prime if and only if the inclusion [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text] implies that [Formula: see text] or [Formula: see text]. Also, [Formula: see text]-absorbing prime ideals of PIDs, valuation domains, Prufer domains and idealization of a modules are characterized. Finally, an analogous to the Prime Avoidance Theorem and some applications of this theorem are given.