Mohammad adarbeh | Searcharxiv

arXiv2025

Uniformly S-pseudo-projective modules

In this paper, we introduce the notion of uniformly S-pseudo-projective (u-S-pseudo-projective) modules as a generalization of u-S-projective modules. Let R be a ring and S a multiplicative subset of R. An R-module P is said to be u-S-pseudo-projective if for any submodule K of P, there is s\in S such that for any u-S-epimorphism f:P\to \frac{P}{K}, sf can be lifted to an endomorphism g:P\to P. We prove that an R-module M is u-S-quasi-projective if and only if M\oplus M is u-S-pseudo-projective. Also, we prove that if A\oplus B is u-S-pseudo-projective, then any u-S-epimorphism f:A\to B u-S-splits. We give characterizations of certain classes of rings, such as u-S-semisimple and strongly S-perfect rings.

math.AC↗

arXiv2025

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

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.

math.AC↗