The concepts have been introduced in Almost DistributiveLattices(ADLs), namely, R-ideals and-ideals. A set ofconditions has been identified that are equivalent to converting anE-ideal into an R-ideal. Moreover, it has been derived that forany E-ideal, there exists a homomorphism with a dual dense kernel,which is itself an R-ideal. The characterization of-ideals interms of R-ideals and congruences has been established. Additionally,equivalent conditions have been established to demonstratethat the space of all prime-ideals forms a Hausdorff space.
We study those modules in which all submodules contain a nonzero submodule with Noetherian dimension less than or equal to $\alpha $ where $\alpha $ is the least ordinal number with this property, calling them modules that have enough $\alpha$-noetherians. Also, we study those modules in which any nonzero factor module contains a nonzero submodule with Noetherian dimension less than or equal to $\alpha$, where $\alpha$ is the least ordinal number with this property, calling them $\alpha$-seminoetherians. Our work extends the results of Kourki and Tribak in Commu. Algebra (2022).
The rings considered in this paper are commutative with identity and the modules considered are modules over commutative rings and are unitary. We use R to denote a ring, S to denote a multiplicatively closed subset of R, and M to denote a module over R. We say that M is an S-Laskerian (respectively, a strongly S-Laskerian) R-module if M is an S-finite R-module and for any submodule N of M, either (N: M) meets S or there exist t in S and an S-decomposable (respectively, a strongly S-decomposable) submodule K of M such that tN is a submodule of K and K is contained in N. We say that M is a weakly S-Laskerian (respectively, weakly strongly S-Laskerian) R-module if each S-finite proper submodule of M is an S-Laskerian (respectively, a strongly Laskerian ) R-module. In this paper, we discuss some results on basic properties of weaklly S-Laskerian modules and we extend some of the properties of S-Laskerian modules to weakly S-Laskerian modules.
In this paper, we consider certain classes of hypergraphs constructed from the tensor join of hypergraphs, specifically the tensor join of hypergraphs constrained by vertex subsets and the $(H, \mathcal{T}_{\mathcal{S}})$-join of hypergraphs constrained by $\mathcal{S}$. We determine some eigenvalues of the adjacency tensor of these classes of hypergraphs by establishing corresponding eigenvectors. We demonstrate that the eigenvalues of the adjacency tensor of the constituting hypergraphs are also eigenvalues of the adjacency tensor of the join of a set of hypergraphs. Furthermore, as a special case of our results, we provide some eigenvalues and eigenvectors of the adjacency tensor for the join of non-uniform hypergraphs on a backbone hypergraph $H$ (and, similarly, for the join of $m$-uniform hypergraphs on a backbone hypergraph $H$). Additionally, we establish a relationship between the eigenvalues of the adjacency tensor of a hypergraph $H$ and certain eigenvalues of the adjacency tensor of the $(H, \mathcal{T}_{\mathcal{S}})$-join of hypergraphs constrained by $\mathcal{S}$. Using this relationship, we determine some eigenvalues and their corresponding eigenvectors for the adjacency tensor of the lexicographic product of two hypergraphs.
In their construction of the Stone-\v{C}ech compactification using only ring ideals (as opposed to $\ell$-ideals), Banaschewski and Sioen define a certain nucleus on the coherent frame $\Rid(A)$ of radical ideals of a commutative ring $A$. In this paper we extend this nucleus to any algebraic frame that has a dense compact element and in which the meet of two compact elements is compact. Of course, not every such algebraic frame is coherent, so the extension is indeed a genuine extension. We then study some properties of this nucleus which are not considered by Banaschewski and Sioen.
