On purely-prime ideals with applications

Flat topology on prime, maximal and minimal prime spectra of quantales

“…We shall obtain new characterizations of these types of quantales in terms of some algebraic and topological properties of spectra. Our results extend some theorems proven in [1], [43], [44], [45], [3], [4], etc. for the spectra of commutative rings.…”

“…Recently, Tarizadeh proposed in [43] purely algebraic definitions for flat and patch topologies on the prime spectrum Spec(R) of a commutative ring R. For examples, the closed sets in the flat topology on Spec(R) are defined as the images Im(f * ), where f * : Spec(A) → Spec(R) is the map induced by a flat morphism f : R → A. Further, the flat and the patch topologies were used to obtain new properties and new characterizations of Gelfand rings and clean rings, as well as of new classes of rings: mp-rings and purified rings (see [1], [44], [45]). A natural problem is to use the flat and the patch topology for obtaining new results on the spectra of other types of algebras.…”

Reticulation of a quantale, pure elements and new transfer properties

“…We mention that most of the results of this paper extend some theorems that appear in the case of commutative rings ( [1], [3], [5], [6], [11], [12], [30], [35], [43], [48], [49]), bounded distributive lattices ( [4], [17], [18], [26], [30], [40], [43], [45]), MV-algebras ( [9], [14]), BL -algebras [32], residuated lattices ( [36], [41]),etc.…”

“…It collects various conditions that characterize normal quantales. In particular, the below properties (1) -( 7) correspond to some conditions from Theorem 4.3 of [1] and the properties ( 8) -( 14) generalize the conditions contained in Theorem 4.6 of [49].…”

“…The literature of ring theory contains several properties that characterize the (commutative) Gelfand rings (see [43], [30], [35], [1], [49]). The following result extends the main characterization theorems of Gelfand rings.…”