On 1-absorbing prime and weakly 1-absorbing prime subsemimodules

Abstract

In this paper, we introduce the concepts of 1-absorbing prime and weakly 1-absorbing prime subsemimodules over commutative semirings. Let S be a commutative semiring with 1 \neq 0 and M an S-semimodule. A proper subsemimodule N of M is called 1-absorbing prime (weakly 1-absorbing prime) if, for all nonunits a, b \in S and m \in M, abm \in N (0 \neq abm \in N) implies ab \in (N :_{S} M) or m \in N. We study many properties of these concepts. For example, we show that a proper subsemimodule N of M is 1-absorbing prime if and only if for all proper ideals I, J of S and subsemimodule K of M with IJK \subseteq N, either IJ \subseteq (N:_{S} M) or K \subseteq N. Also, we prove that a proper subtractive subsemimodule N of M is weakly 1-absorbing prime if and only if for all proper ideals I, J of S and subsemimodule K of M with 0 \neq IJK \subseteq N, either IJ \subseteq (N:_{S} M) or K \subseteq N.