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Let 픔 be a reduced left-sided and left-stable quantale without zero divisors. This paper shows how the systematic use of the «quantization» of 2 viewed as a regular and strict right 픔-subalgebra of 픔 can both strengthen and simplify the rather complicated construction of quantale-enriched topologies given for semi-unital and semi-integral quantales in the original paper [3]. The underlying structure for this new construction is an adjoint situation between the dual category of semi-unital, semi-integral quantales and quantized topological spaces. [ABSTRACT FROM AUTHOR]

Let O denote the ring of integers of a quadratic field Q ( − 7 ) . In 2022, Murtuza and Garge [Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian Journal of Pure and Applied Mathematics, Published online; December 2022.] gave a necessary and sufficient condition for a diagonal quadratic form a 1 X 1 2 + a 2 X 2 2 + a 3 X 3 2 where a i ∈ O for 1 ≤ i ≤ 3 for representing all 2 × 2 matrices over O . Let K denote a quadratic field such that its ring of integers O K is a principal ideal domain and 2 is a product of two distinct primes. It is a well-known fact that Q ( − 7 ) is the only imaginary quadratic field with the above properties. Let D K denote the discriminant of K. We have D K ≡ 1 ( mod  8 ) if and only if 2 is a product of two distinct primes in O K . With O K as above, in this paper we generalize our earlier result. We give a necessary and sufficient condition for a diagonal quadratic form ∑ i = 1 m a i X i 2 where a i ∈ O K , 1 ≤ i ≤ m to represent all 2 × 2 matrices over O K . This result is a conjecture stated in [Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian Journal of Pure and Applied Mathematics, Published online; December 2022]. [ABSTRACT FROM AUTHOR]

Let L be a C-lattice and M be a lattice module over L. For N ∊ M, meet of all prime elements K of M with (N: 1M) ≥ (K: 1M) is called the Zariski prime radical of N and is denoted by zp ✓N. In this paper, we study the concept of the Zariski prime radical of elements of M and investigate their properties. [ABSTRACT FROM AUTHOR]

Let M be a lattice module over a C-lattice L. The maximal spectrum Max(M) of M is the collection of all maximal elements of M. In this paper, we study the topology on Max(M) and also the interrelations between the topological properties of Max(M) and the algebraic properties of M are established. [ABSTRACT FROM AUTHOR]

Let K be an imaginary quadratic field whose ring of integers OK is a Euclidean domain. In our earlier work, the following four results are proved. First, for β 2 OK\{0}, each element in OK has a base β-expansion whose digits are bounded by certain constants. Secondly, if π = α nβn + α n-1βn-1 + · · · + α 1β + α0 := f(β) is a base β-expansion of a prime in OK, and the digits α n and α n-1 satisfy some natural restrictions, then the polynomial f(x) is irreducible over K. Thirdly, for Gaussian integers, similar base β-expansion but with digits belonging to a complete residue system modulo β is also valid. Lastly, irreducibility results similar to that in the second result continue to hold for Gaussian integers with base expansion as described in the third result. In this paper, we establish some generalizations of the above irreducibility results by considering ωπ (ωϵOK) instead of π. [ABSTRACT FROM AUTHOR]

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 SpecpLq 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 MaxpLq consisting of all maximal elements of L. [ABSTRACT FROM AUTHOR]

Let L be a multiplicative lattice and M be an L-module. In this study, we present a topology said to be the Zariski topology over σ(M), the collection of all prime elements of an L-module M. We research some results on the Zariski topology over σ(M). We show that the topology is a T0 -space and a T1 -space under some conditions. Some properties and results are studied for the topology over σ(L) , the collection of all prime elements of a multiplicative lattice L. [ABSTRACT FROM AUTHOR]

The second spectrum Specs(M) is the collection of all second elements of M. In this paper, we study the topology on Specs(M), which is a generalization of the Zariski topology on the prime spectrum of lattice modules. Besides some properties, Specs(M) is characterized and the interrelations between the topological properties of Specs(M) and the algebraic properties of M, are studied. [ABSTRACT FROM AUTHOR]

In this paper, we find the ω-value of the generators of any numerical semigroup with embedding dimension three. This allows us to determine all possible orderings of the ω-values of the generators. In addition, we relate the ω-value of the numerical semigroup to its catenary degree. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we introduce the pointfree version of rough sets. For this we consider a lattice L instead of the power set P(X) of a set X. We study the properties of lower and upper pointfree approximation, precise elements, and their relation with prime elements. Also, we study lower and upper pointfree approximation as a Galois connection, and discuss the relations between partitions and Galois connections. [ABSTRACT FROM AUTHOR]

Let L be a compactly generated multiplicative lattice with 1 compact in which every finite product of compact elements is compact and M be a module over L. In this paper we generalize the concepts of Baer elements,*-elements and closed elements and obtain the relation between *-elements and Baer elements and also closed elements and Baer elements. Some characterization are also obtain for closed elements of M and minimal prime elements of M. [ABSTRACT FROM AUTHOR]

Prime elements of right-sided and idempotent quantales are identified with strong homomorphisms with values in the non-commutative, idempotent and right-sided chain with three elements. On this basis an adjunction between the category of right-sided idempotent quantales and the category of three-valued topological spaces is established. There is a second adjunction based on quantum spaces which is an extension of the duality between spatial right-sided idempotent quantales and sober quantum spaces. Both adjunctions restricts to the well known adjunction between topological spaces and locales. In particular, quantum spaces form a full subcategory of the category of three-valued topological spaces. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of classical primary and classical quasi primary elements in lattice modules which are the generalization of the concepts in submodules. We obtain some characterizations of classical primary and classical quasi primary elements .We also investigate the decomposition and minimal decomposition into classical quasi primary elements. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of n-Baer lattice and Mn-normal lattice with respect to a radical element u. It is shown that a quasi-regular lattice L is n-Baer if and only if L is Mn-normal. [ABSTRACT FROM AUTHOR]

A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded semiring are studied. In particular, it is proved that under some mild assumption, the set Z( A) of nonzero zero divisors of A is A \ {0, 1}, and each prime element of A is a maximal element. For a bounded semiring A with Z( A) = A \ {0, 1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either | Z( A)| = 1 or | Z( A)| = 2 and Z( A) ≠ 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset II( R) is pure and shellable, where II( R) consists of all ideals of R. [ABSTRACT FROM AUTHOR]

In this paper, we study primary decomposition of elements in lattice modules. A necessary and sufficient condition for a prime element p of a multiplicative lattice L to be equal to some associated prime of an element in a lattice module having primary decomposition is obtained. [ABSTRACT FROM AUTHOR]

Let D be an integral domain. In this paper, we investigate two (integer- or ∞-valued) invariants ω(D, x) and ω(D) which measure how far a nonzero x ∈ D is from being prime and how far an atomic integral domain D is from being a unique factorization domain (UFD), respectively. In particular, we are interested in when there is a nonzero (irreducible) x ∈ D with ω(D, x) = ∞ and the relationship between ω(A, x) and ω(B, x), and ω(A) and ω(B), for an extension A ⊆ B of integral domains and a nonzero x ∈ A. [ABSTRACT FROM AUTHOR]

Let D be an integral domain. We investigate two invariants ω(D, x) and ω(D) which measure how far an x ∈ D is from being prime and how far an atomic integral domain D is from being a UFD, respectively. We give a new characterization of number fields with class number two. We also study asymptotic versions of these two invariants. [ABSTRACT FROM AUTHOR]

Abstract: A graph monoid is a commutative monoid for which there is a particularly simple presentation, given in terms of a quiver. Such monoids are known to satisfy various nonstable K-theoretical representability properties for either von Neumann regular rings or -algebras. We give a characterization of graph monoids within finitely generated antisymmetric refinement monoids. This characterization is formulated in terms of the prime elements of the monoid, and it says that each free prime has at most one free lower cover. We also characterize antisymmetric graph monoids of finite quivers. In particular, the monoid is a graph monoid, but it is not the graph monoid of any finite quiver. [Copyright &y& Elsevier]

Primeness on modules can be defined by prime elements in a suitable partially ordered groupoid. Using a product on the lattice of submodules ℒ(M) of a module M defined in [3] we revise the concept of prime modules in this sense. Those modules M for which ℒ(M) has no nilpotent elements have been studied by Jirasko and they coincide with Zelmanowitz' "weakly compressible" modules. In particular we are interested in representing weakly compressible modules as a subdirect product of "prime" modules in a suitable sense. It turns out that any weakly compressible module is a subdirect product of prime modules (in the sense of Kaplansky). Moreover if M is a self-projective module, then M is weakly compressible if and only if it is a subdirect product of prime modules (in the sense of Bican et al.). An application to Hopf actions is given. [ABSTRACT FROM AUTHOR]

This paper presents the topologization of semi-unital and semi-integral quantales by means of enriched topologies. In a first step we show that semi-unital and semi-integral quantales can be represented by a specific type of right Q 2 ˆ -algebras in Sup where Q 2 ˆ is the unitalization of the quantization of 2. In a second step we use this identification and construct the corresponding Q 2 ˆ -enriched sober spaces. On this basis semi-unital and spatial quantales are characterized by Q 2 ˆ -enriched topologies. A consequence of these constructions is a natural topologization of the quantale of all closed left (resp. right) ideals of a non-commutative and unital C ⁎ -algebra. [ABSTRACT FROM AUTHOR]