On generalization of prime, weakly prime and almost prime elements in multiplicative lattices

2014 ◽  
Vol 8 ◽  
pp. 439-449 ◽  
Author(s):  
Sachin Ballal ◽  
Vilas Kharat
Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
C. S. Manjarekar ◽  
A. V. Bingi

We investigate ϕ-prime and ϕ-primary elements in a compactly generated multiplicative lattice L. By a counterexample, it is shown that a ϕ-primary element in L need not be primary. Some characterizations of ϕ-primary and ϕ-prime elements in L are obtained. Finally, some results for almost prime and almost primary elements in L with characterizations are obtained.


2016 ◽  
Vol 14 (1) ◽  
pp. 673-680
Author(s):  
Emel Aslankarayigit Ugurlu ◽  
Fethi Callialp ◽  
Unsal Tekir

AbstractIn this paper, we study multiplication lattice modules. We establish a new multiplication over elements of a multiplication lattice module.With this multiplication, we characterize idempotent element, prime element, weakly prime element and almost prime element in multiplication lattice modules.


2016 ◽  
Vol 09 (04) ◽  
pp. 1650071
Author(s):  
Vinayak Joshi ◽  
Sachin Sarode

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].


1995 ◽  
Vol 31 (3) ◽  
pp. 201-208 ◽  
Author(s):  
C. Jayaram ◽  
E. W. Johnson

2015 ◽  
Vol 48 (4) ◽  
Author(s):  
Y.T. Nai ◽  
D. Zhao

AbstractWe 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.


2015 ◽  
Vol 08 (04) ◽  
pp. 1550066 ◽  
Author(s):  
Sachin Ballal ◽  
Vilas Kharat

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].


2012 ◽  
Vol 40 (8) ◽  
pp. 2825-2840 ◽  
Author(s):  
Fethi Çallıalp ◽  
C. Jayaram ◽  
Ünsal Tekir

Author(s):  
Christopher Tomlins

As the linguistic/cultural turn of the last fifty years has begun to ebb, sociolegal and legal-humanist scholarship has seen an accelerating return to materiality. This chapter asks what relationship may be forthcoming between the “new materialisms” and “vibrant matter” of recent years, and the older materialisms—both historical and literary, both Marxist and non-Marxist—that held sway prior to post-structuralism. What impact might such a relationship have on the forms, notably “spatial justice,” that materiality is assuming in contemporary legal studies? To attempt answers, the chapter turns to two figures from more than half a century ago: Gaston Bachelard—once famous, now mostly forgotten; and Walter Benjamin—once largely forgotten, now famous. A prolific and much-admired writer between 1930 and 1960, Bachelard pursued two trajectories of inquiry: a dialectical and materialist and historical (but non-Marxist) philosophy of science; and a poetics of the material imagination based on inquiry into the literary reception and representation of the prime elements—earth, water, fire, and air. Between the late 1920s and 1940, meanwhile, Benjamin developed an idiosyncratic but potent form of historical materialism dedicated to “arousing [the world] from its dream of itself.” The chapter argues that by mobilizing Bachelard and Benjamin for scholarship at the intersection of law and the humanities, old and new materialisms can be brought into a satisfying conjunction that simultaneously offers a poetics for spatial justice and lays a foundation for a materialist legal historiography for the twenty-first century.


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