2025/09/18 by adarbeh, Mohammad, Saleh, Mohammad
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.15398
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 ≠ 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 ∈ S and m ∈ M, abm ∈ N (0 ≠ abm ∈ N) implies ab ∈ (N :S M) or m ∈ 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 ⊆ N, either IJ ⊆ (N:S M) or K ⊆ 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 ≠ IJK ⊆ N, either IJ ⊆ (N:S M) or K ⊆ N.