Diameter and girth of the multiplicative zero-divisor graph of multiplicative lattices

In this paper, we study the multiplicative zero-divisor graph [Formula: see text] of a multiplicative lattice [Formula: see text]. Under certain conditions, we prove that for a reduced multiplicative lattice [Formula: see text] having more than two minimal prime elements, [Formula: see text] contains a cycle and [Formula: see text]. This essentially settles the conjecture of Behboodi and Rakeei [The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10(4) (2011) 741–753]. Further, we have characterized the diameter of [Formula: see text].

A GCD and LCM-like inequality for multiplicative lattices

Let $A_1,\ldots,A_n$ $(n\ge 2)$ be elements of an commutative multiplicative lattice. Let $G(k)$ (resp., $L(k)$) denote the product of all the joins (resp., meets) of $k$ of the elements. Then we show that $$L(n)G(2)G(4)\cdots G(2\lfloor n/2 \rfloor ) \leq G(1)G(3)\cdots G(2\lceil n/2 \rceil -1).$$ In particular this holds for the lattice of ideals of a commutative ring. We also consider the relationship between $$G(n)L(2)L(4)\cdots L(2\lfloor n/2 \rfloor ) \text{ and } L(1)L(3)\cdots L(2\lceil n/2 \rceil -1)$$and show that any inequality relationships are possible.

Open Set Lattices of Subspaces of Spectrum Spaces

We take a unified approach to study the open set lattices of various subspaces of the spectrum of a multiplicative lattice L. The main aim is to establish the order isomorphism between the open set lattice of the respective subspace and a sub-poset of L. The motivating result is the well known fact that the topology of the spectrum of a commutative ring R with identity is isomorphic to the lattice of all radical ideals of R. The main results are as follows: (i) for a given nonempty set S of prime elements of a multiplicative lattice L, we define the S-semiprime elements and prove that the open set lattice of the subspace S of Spec(L) is isomorphic to the lattice of all S-semiprime elements of L; (ii) if L is a continuous lattice, then the open set lattice of the prime spectrum of L is isomorphic to the lattice of all m-semiprime elements of L; (iii) we define the pure elements, a generalization of the notion of pure ideals in a multiplicative lattice and prove that for certain types of multiplicative lattices, the sub-poset of pure elements of L is isomorphic to the open set lattice of the subspace Max(L) consisting of all maximal elements of L.

Zariski topology on lattice modules

Let [Formula: see text] be a lattice module over a [Formula: see text]-lattice [Formula: see text] and [Formula: see text] be the set of all prime elements in lattice modules [Formula: see text]. In this paper, we study the generalization of the Zariski topology of multiplicative lattices [N. K. Thakare, C. S. Manjarekar and S. Maeda, Abstract spectral theory II: Minimal characters and minimal spectrums of multiplicative lattices, Acta Sci. Math. 52 (1988) 53–67; N. K. Thakare and C. S. Manjarekar, Abstract spectral theory: Multiplicative lattices in which every character is contained in a unique maximal character, in Algebra and Its Applications (Marcel Dekker, New York, 1984), pp. 265–276.] to lattice modules. Also we investigate the interplay between the topological properties of [Formula: see text] and algebraic properties of [Formula: see text].

On ϕ-Absorbing Primary Elements in Lattice Modules

Let L be a C-lattice and let M be a lattice module over L. Let ϕ:M→M be a function. A proper element P∈M is said to be ϕ-absorbing primary if, for x1,x2,…,xn∈L and N∈M, x1x2⋯xnN≤P and x1x2⋯xnN≰ϕ(P) together imply x1x2⋯xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤PM, for some i∈{1,2,…,n}. We study some basic properties of ϕ-absorbing primary elements. Also, various generalizations of prime and primary elements in multiplicative lattices and lattice modules as ϕ-absorbing elements and ϕ-absorbing primary elements are unified.