Let $R$ be a commutative ring with identity and $M$ be an $R$-module. It is shown that the usual Lattice $\mathcal{V}(_{R}M)$ of varieties of submodules of $M$ is a distributive Lattice. If $M$ is a semisimple $R$-module and the unary operation $^{\prime}$ on $\mathcal{V}(_{R}M)$ is defined by $(V(N))^{\prime}=V(\tilde{N})$, where $M=N\oplus \tilde{N}$, then the Lattice $\mathcal{V}(_{R}M)$ with $^{\prime}$ forms a Boolean algebra. In this paper, we examine the properties of certain mappings between $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$, in particular considering when these mappings are Lattice homomorphisms. It is shown that if $M$ is a faithful primeful $R$-module, then $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$ are isomorphic Lattices, and therefore $\mathcal{V}(_{R}M)$ and the Lattice $\mathcal{R}(R)$ of radical ideals of $R$ are anti-isomorphic Lattices. Moreover, if $R$ is a semisimple ring, then $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$ are isomorphic Boolean algebras, and therefore $\mathcal{V}(_{R}M)$ and $\mathcal{L}(R)$ are anti-isomorphic Boolean algebras.

Let G be a simple, oriented connected graph with n vertices and m edges. Let I (B) be the binomial ideal associated to the incidence matrix B of the graph G. Assume that IL is the Lattice ideal associated to the rows of the matrix B. Also let Bi be a sub matrix of B after removing the i -th row. We introduce a graph theoretical criterion for G which is a sufficient and necessary condition for I (B) =I (Bi) and I (Bi) =IL. After that we introduce another graph theoretical criterion for G which is a sufficient and necessary condition for I (B) =IL. It is shown that the heights of I (B) and I (Bi) are equal to n - 1 and the dimensions of I (B) and I (Bi) are equal to m - n+1; then I (Bi) is a complete intersection ideal.

Let M be a Lattice module over the multiplicative Lattice L. An L-module M is called a multiplication Lattice module if for every element NÎM there exists an element aÎL such that N=a1M. Our objective is to investigate properties of prime elements of multiplication Lattice modules.

in this work, some of the Lattice plasmon quantum features are examined. Initially, the interaction of the far-field photonic mode and the nanoparticle plasmon mode is investigated. We probe the optical properties of the array plasmon that are dramatically affected by the array geometry. It is notable to mention that the original goal of this work is to examine the quantum feature of the array plasmon. For this reason, we consider a system containing array of the plasmonic nanoparticles and quantum dots. For a complete understanding, we analyze the system with the full quantum theory. Notably, the full quantum analyzing enables us to investigate the quantum fluctuation of the array field. Here, for instance, we study the second-order correlation function and report its modeling results.

This paper addresses the need for an efficient and adaptable approach to solve linear acoustic wave equations in the Lattice Boltzmann method (LBM). A novel Lattice-adaptive model is introduced, derived through a Chapman–Enskog analysis, which utilizes a single relationship for the equilibrium distribution function across all Lattice structures. The intended derivation begins by considering a standard equilibrium distribution function with unknown coefficients. By selecting the displacement of the acoustic wave as the zero-order microscopic moment, accurate recovery of the macroscopic wave equation is ensured. Unlike existing methods, the model simplifies the complexity associated with equilibrium distribution functions and offers greater versatility. The model is validated through extensive benchmark testing on one and two-dimensional wave propagation problems. Results demonstrate excellent agreement with analytical solutions, with maximum root mean square errors of 10-3 (<0.1% error) and minimum errors of 10-6 (<0.0001% error), indicating high predictive accuracy (>99.9%). Additionally, the model exhibits second-order spatial accuracy, with the relative error norm E_2 displaying slopes close to 2, signifying a spatial accuracy of second order. The numerical simulations show a decrease in errors as the mesh size becomes more refined.

We show that the Lattice of all ideals of a ring R can be embedded in the Lattice of all its fuzzy ideals in uncountably many ways. For this purpose, we introduce the concept of the generalized characteristic function Xrs(A) of a subset A of a ring R for fixed r, s Î [0,1] and show that A is an ideal of R if, and only if, its generalized characteristic function Xrs (A) is a fuzzy ideal of R. We also show that the set of all generalized characteristic functions Crs(I(R)) of the members of I(R) for fixed r, s Î [0,1] is a complete subLattice of the Lattice of all fuzzy ideals of R and establish that this latter Lattice is generated by the union of all its complete sub Lattices Crs (I(R)).