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For a finite simple undirected graph G , the universal adjacency matrix U (G) is a linear combination of the adjacency matrix A (G) , the degree diagonal matrix D (G) , the identity matrix I and the all-ones matrix J , that is U (G) = α A (G) + β D (G) + γ I + η J , where α , β , γ , η ∈ ℝ and α ≠ 0. The cozero-divisor graph Γ ′ (R) of a finite commutative ring R with unity is a simple undirected graph with the set of all nonzero nonunits of R as vertices and two vertices x and y are adjacent if and only if x ∉ y R and y ∉ x R. In this paper, we study structural properties of Γ ′ (R) by defining an equivalence relation on its vertex set in terms of principal ideals of the ring R. Then we obtain the universal adjacency eigenpairs of Γ ′ (R) and its complement, and as a consequence one may obtain several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree diagonal matrix of Γ ′ (R) and Γ ′ (R) ¯ in an unified way. Moreover, we get the universal adjacency eigenpairs of the cozero-divisor graph and its complement for a reduced ring and the ring of integers modulo m in a simpler form. [ABSTRACT FROM AUTHOR]

For each commutative ring R we associate a simple graph Γ1∗(R). We investigate the properties of an Artinian ring R when Γ1∗(R) is a star. [ABSTRACT FROM AUTHOR]

Two forms f (x, y, z) and g(x, y, z) of degree 3 were constructed, with their values being the norms of numbers in the subfields of degree 3 of the circular fields K13 and K19 , respectively. Using the decomposition law in a circular field, Diophantine equations f (x, y, z) = a and g(x, y, z) = b , where a, b ∈ Z, a 6= 0, b 6= 0 were solved. The assertions that, based on the canonical decomposition of the numbers a and b into prime factors, make it possible to determine whether the equations f (x, y, z) = a and g(x, y, z) = b have solutions were proved.

We define a semi-Brouwerian almost distributive lattice as a description of a semi-Brouwerian algebra in the category of almost distributive lattices and derive a number of algebraic properties on them. We obtain a few equivalent conditions for an almost distributive lattice to become a semi-Brouwerian almost distributive lattice. Finally we identify a class of pseudo-complemented lattices and a class of semi-Brouwerian algebras in a semi-Brouwerian almost distributive lattice. [ABSTRACT FROM AUTHOR]

Several results about one-sided duo rings and duo rings are generalized from the case of unital rings to the case of arbitrary associative rings in this paper. For example, the characterization of one-sided duo rings and duo rings is given, and it is shown that all idempotents of a duo ring commute with all elements of the ring and that every prime ideal is completely prime.

Let 풪풫ℰn be the monoid of all orientation-preserving and extensive full transformations on {1,…,n}. In this paper, first we discuss left and right abundance of the principal ideals of 풪풫ℰn. Second, we give necessary and sufficient conditions for the principal ideals of 풪풫ℰn to be abundant and regular. Finally, we study abundance of the ideals in 풪풫ℰn. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a few properties of w -invertible ideals over infra-Krull domains. Let w - ℐ (R) denote the multiplicative group of w -invertible w -fractional ideals of an integral domain R and let (R) denote the multiplicative group of principal fractional ideals of an integral domain R. We proved that, if R is an infra-Krull domain, then w - ℐ (R) ≅ ⊕ P ∈ w - Max (R) (R P). As an application, we can get one of Neukirch's results directly. [ABSTRACT FROM AUTHOR]

We study a new family of metrics, weighted poset block metric, which generalizes the weighted coordinates poset metric introduced by Panek and Pinheiro (IEEE Trans Inf Theory 66(11):6823–6834, 2020) and the metric for linear error-block codes introduced by Feng et al. (Finite Fields Appl 12(4):638–652, 2006). This metric covers various metrics such as Hamming metric, Lee metric, poset metric, pomset metric, poset block metric, pomset block metric and so on. We give a complete description of the groups of linear isometries of these metric spaces in terms of a semi-direct product. Moreover, we obtain a Singleton type bound for codes equipped with weighted poset block metric and define MDS codes. When the poset is a chain, we study the relationship between MDS codes and perfect codes. [ABSTRACT FROM AUTHOR]

Several results about one-sided duo rings and duo rings are generalized from the case of unital rings to the case of arbitrary associative rings in this paper. For example, the characterization of one-sided duo rings and duo rings is given, and it is shown that all idempotents of a duo ring commute with all elements of the ring and that every prime ideal is completely prime. [ABSTRACT FROM AUTHOR]

We prove that the grades of simple modules indexed by boolean permutations, over the incidence algebra of the symmetric group with respect to the Bruhat order, are given by Lusztig's $\mathbf{a}$-function. Our arguments are combinatorial, and include a description of the intersection of two principal order ideals when at least one permutation is boolean. An important object in our work is a reduced word written as minimally many runs of consecutive integers, and one step of our argument shows that this minimal quantity is equal to the length of the second row in the permutation's shape under the Robinson-Schensted correspondence. We also prove that a simple module over the above-mentioned incidence algebra is perfect if and only if its index is the longest element of a parabolic subgroup.Mathematics Subject Classifications: 20F55, 06A07, 05E15

Let C n denote the order-preserving and decreasing full transformation semigroup on X n = { 1 , 2 , ... , n } . In this paper, first we describe each principal ideal of C n . Second, we discuss left and right abundance of the principal ideals of C n . Further, necessary and sufficient conditions for the principal ideals in C n to be abundant and regular are given. Finally, we study abundance of the ideals in C n . [ABSTRACT FROM AUTHOR]

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Characterizations of α-prime ideals in the semiring of nonnegative integers are investigated. [ABSTRACT FROM AUTHOR]

This paper deals with the number theoretic properties of non-unit elements of the ring Zn. Let D be the set of all non-trivial divisors of a positive integer n. Let D1 and D2 be the subsets of D having the least common multiple which are incongruent to zero modulo n with every other element of D and congruent to zero modulo n with at least one another element of D, respectively. Then D can be written as the disjoint union of D1 and D2 in Zn. We explore the results on these sets based on all the characterizations of n. We obtain a formula for enumerating the cardinality of the set of all non-unit elements in Zn whose principal ideals are equal. Further, we present an algorithm for enumerating these sets of all non-unit elements.

For a positive integer d, a non-negative integer n and a non-negative integer h ≤ n , we study the number C n (d) of principal ideals; and the number C n , h (d) of principal ideals generated by an element of rank h, in the d-tonal partition monoid on n elements. We compute closed forms for the first family, as partial cumulative sums of known sequences. The second gives an infinite family of new integral sequences. We discuss their connections to certain integral lattices as well as to combinatorics of partitions. [ABSTRACT FROM AUTHOR]

We study the existence of Banach algebra structures on simple extensions of a unitary commutative Banach algebra. The link with the integrality of these extensions is studied. For any simple extension, a characterization of the existence of the Banach algebra norm making continuous the canonical injection is also given. [ABSTRACT FROM AUTHOR]

The paper studies Rogers semilattices for families of equivalence relations in the Ershov hierarchy. For an arbitrary notation a of a nonzero computable ordinal, we consider ∑ a − 1 -computable numberings of the family of all ∑ a − 1 equivalence relations. We show that this family has infinitely many pairwise incomparable Friedberg numberings and infinitely many pairwise incomparable positive undecidable numberings. We prove that the family of all c.e. equivalence relations has infinitely many pairwise incomparable minimal nonpositive numberings. Moreover, we show that there are infinitely many principal ideals without minimal numberings. [ABSTRACT FROM AUTHOR]

C∗-subalgebra of the algebra of all bounded operators on the Hilbert space l2 generated by the multiplier algebra and a family of partial isometries satisfying some relations is covered in the paper. The ideals of the algebra under study, as well as the ideals of the quotient algebra, are considered over compact operators. It is demonstrated that the quotient algebra can be represented as a direct sum of two principal ideals and has nontrivial center.

We describe the periodic groups whose endomorphism rings satisfy the annihilator condition for the principal left ideals generated by nilpotent elements. We prove that torsion-free reduced separable, vector, and algebraically compact groups have endomorphism rings with the annihilator condition for the principal left (right) ideals generated by nilpotent elements if and only if these rings are commutative. We show that the almost injective groups (in the sense of Harada) are injective, i.e. divisible. [ABSTRACT FROM AUTHOR]