scholarly journals ϕ-Prime and ϕ-Primary Elements in Multiplicative Lattices

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
C. S. Manjarekar ◽  
A. V. Bingi
Keyword(s):  
Multiplicative Lattice ◽  
Primary Element ◽  
Prime Elements ◽  
Compactly Generated ◽  
Almost Prime ◽  
Multiplicative Lattices

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.

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

2016 ◽  
Vol 09 (04) ◽  
pp. 1650071
Author(s):  
Vinayak Joshi ◽  
Sachin Sarode
Keyword(s):  
Zero Divisor ◽  
Commutative Rings ◽  
Zero Divisor Graph ◽  
Multiplicative Lattice ◽  
Minimal Prime ◽  
Prime Elements ◽  
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].

scholarly journals Open Set Lattices of Subspaces of Spectrum Spaces

2015 ◽  
Vol 48 (4) ◽  
Author(s):  
Y.T. Nai ◽  
D. Zhao
Keyword(s):  
Continuous Lattice ◽  
Prime Spectrum ◽  
Unified Approach ◽  
Maximal Elements ◽  
Order Isomorphism ◽  
Open Set ◽  
Multiplicative Lattice ◽  
Radical Ideals ◽  
Prime Elements ◽  
Multiplicative Lattices

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.

scholarly journals Prime, weakly prime and almost prime elements in multiplication lattice modules

2016 ◽  
Vol 14 (1) ◽  
pp. 673-680
Author(s):  
Emel Aslankarayigit Ugurlu ◽  
Fethi Callialp ◽  
Unsal Tekir
Keyword(s):  
Prime Element ◽  
Idempotent Element ◽  
Prime Elements ◽  
Almost Prime

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.

scholarly journals Residuation Properties and Weakly Primary Elements in Lattice Modules

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
C. S. Manjarekar ◽  
U. N. Kandale
Keyword(s):  
Prime Element ◽  
Multiplicative Lattice ◽  
Primary Element

We obtain some elementary residuation properties in lattice modules and obtain a relation between a weakly primary element in a lattice module M and weakly prime element of a multiplicative lattice L.

scholarly journals Generalized prime elements in a compactly generated $l$-semigroup, II

1973 ◽  
Vol 49 (5) ◽  
pp. 310-313
Author(s):  
Kentaro Murata ◽  
Derbiau F. Hsu
Keyword(s):  
Prime Elements ◽  
Compactly Generated

scholarly journals Generalized prime elements in a compactly generated $l$-semigroup, I

1973 ◽  
Vol 49 (2) ◽  
pp. 134-139
Author(s):  
Kentaro Murata ◽  
Derbiau F. Hsu
Keyword(s):  
Prime Elements ◽  
Compactly Generated

s-Prime elements in multiplicative lattices

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

Zariski topology on lattice modules

2015 ◽  
Vol 08 (04) ◽  
pp. 1550066 ◽  
Author(s):  
Sachin Ballal ◽  
Vilas Kharat
Keyword(s):  
New York ◽  
Spectral Theory ◽  
Zariski Topology ◽  
Topological Properties ◽  
Algebraic Properties ◽  
Prime Elements ◽  
Multiplicative Lattices

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].

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