Some Results on Noetherian Warfield Domains

Let [Formula: see text] be a domain. In this paper, we show that if [Formula: see text] is one-dimensional, then [Formula: see text] is a Noetherian Warfield domain if and only if every maximal ideal of [Formula: see text] is 2-generated and for every maximal ideal[Formula: see text] of [Formula: see text], [Formula: see text] is divisorial in the ring [Formula: see text]. We also prove that a Noetherian domain [Formula: see text] is a Noetherian Warfield domain if and only if for every maximal ideal [Formula: see text] of [Formula: see text], [Formula: see text] can be generated by two elements. Finally, we give a sufficient condition under which all ideals of [Formula: see text] are strongly Gorenstein projective.

Rings with S-acc on d-annihilators

A commutative ring [Formula: see text] is said to satisfy acc on d-annihilators if for every sequence [Formula: see text] of elements of [Formula: see text] the sequence [Formula: see text] is stationary. In this paper we extend the notion of rings with acc on d-annihilators by introducing the concept of rings with [Formula: see text]-acc on d-annihilators, where [Formula: see text] is a multiplicative set. Let [Formula: see text] be a commutative ring and [Formula: see text] a multiplicative subset of [Formula: see text] We say that [Formula: see text] satisfies [Formula: see text]-acc on d-annihilators if for every sequence [Formula: see text] of elements of [Formula: see text] the sequence [Formula: see text] is [Formula: see text]-stationary, that is, there exist a positive integer [Formula: see text] and an [Formula: see text] such that for each [Formula: see text] [Formula: see text] We give equivalent conditions for the power series (respectively, polynomial) ring over an Armendariz ring to satisfy [Formula: see text]-acc on d-annihilators. We also study serval properties of rings satisfying [Formula: see text]-acc on d-annihilators. The concept of the amalgamated duplication of [Formula: see text] along an ideal [Formula: see text] [Formula: see text] is studied.

A study of a family of monomial ideals

In this paper, we study a family of rational monomial parametrizations. We investigate a few structural properties related to the corresponding monomial ideal [Formula: see text] generated by the parametrization. We first find the implicit equation of the closure of the image of the parametrization. Then we provide a minimal graded free resolution of the monomial ideal [Formula: see text], and describe the minimal graded free resolution of the symmetric algebra of [Formula: see text]. Finally, we provide a method to compute the defining equations of the Rees algebra of [Formula: see text] using three moving planes that follow the parametrization.

Extended Green’s relations on Mn(D), a characterization

In this paper, we consider the semiring [Formula: see text] of all [Formula: see text] matrices over a distributive lattice [Formula: see text] and extended Green’s relations [Formula: see text] and [Formula: see text] using [Formula: see text]-ideals. A (left, right) ideal [Formula: see text] of a semiring [Formula: see text] is called a (left, right) [Formula: see text]-ideal if [Formula: see text], where [Formula: see text]. We define [Formula: see text] and [Formula: see text] on a [Formula: see text]-regular semiring [Formula: see text], in which [Formula: see text] is a semilattice, as follows: [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the left [Formula: see text]-ideal generated by [Formula: see text] and [Formula: see text] is the right [Formula: see text]-ideal generated by [Formula: see text]. Here we characterize [Formula: see text] and [Formula: see text] in [Formula: see text] in terms of rows and columns of the matrices.

Formulasi dan Evaluasi Sediaan Obat Kumur (Mouthwash) dari Ekstrak Etanol Daun Kalangkala (Litsea angulata) sebagai Antiseptik Mulut

Research of mouthwash formulation and evaluation of kalangkala leaves extract (Litsea angulata) as an oral antiseptic also has been tested in Streptococcus mutans bacteria by variation in the concentration of extract 2%, 2,5%, and 3%, which aims to determine the ideal formula in physical quality and has the highest antibacterial activity based on an inhibition zone of Streptococcus mutans bacteria in mouthwash formula. The methods used include the step of extraction of Kalangkala leaves, preparation of mouthwash with 3 formula F1 (2%), F2 (2,5%), and F3 (3%) followed by an evaluation that includes organoleptic, pH test, viscosity test, and the test of inhibition zone bacteria. The test of inhibition zone bacteria used NA media with diffusion method. Stability was performed on weeks 1, 2, 3, and 4. The result showed variation in the concentration of extract kalangkala leaves in a mouthwash formula affects the diameter of the inhibition zone. But did not have a significant effect on the physical stability properties of the mouthwash formula. Mouthwash formula has the highest antibacterial activity based on the inhibition zone of Streptococcus mutans that is 3% contained in formula F3.

Hamiltonian trace graph of matrices

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be a positive integer and [Formula: see text] be the set of all [Formula: see text] matrices over [Formula: see text] For a matrix [Formula: see text] Tr[Formula: see text] is the trace of [Formula: see text] The trace graph of the matrix ring [Formula: see text] denoted by [Formula: see text] is the simple undirected graph with vertex set [Formula: see text][Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if Tr[Formula: see text] The ideal-based trace graph of the matrix ring [Formula: see text] with respect to an ideal [Formula: see text] of [Formula: see text] denoted by [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if Tr[Formula: see text] In this paper, we investigate some properties and structure of [Formula: see text] Further, it is proved that both [Formula: see text] and [Formula: see text] are Hamiltonian.

A Construction of Sequentially Cohen–Macaulay Graphs

For every simple graph [Formula: see text], a class of multiple clique cluster-whiskered graphs [Formula: see text] is introduced, and it is shown that all such graphs are vertex decomposable; thus, the independence simplicial complex [Formula: see text] is sequentially Cohen–Macaulay. The properties of the graphs [Formula: see text] and [Formula: see text] constructed by Cook and Nagel are studied, including the enumeration of facets of the complex [Formula: see text] and the calculation of Betti numbers of the cover ideal [Formula: see text]. We also prove that the complex[Formula: see text] is strongly shellable and pure for either a Boolean graph [Formula: see text] or the full clique-whiskered graph [Formula: see text] of [Formula: see text], which is obtained by adding a whisker to each vertex of [Formula: see text]. This implies that both the facet ideal [Formula: see text] and the cover ideal [Formula: see text] have linear quotients.

Note on strongly quasi-primary ideals

Let [Formula: see text] be a commutative ring with [Formula: see text]. A proper ideal [Formula: see text] of [Formula: see text] is said to be a strongly quasi-primary ideal if, whenever [Formula: see text] with [Formula: see text], then either [Formula: see text] or [Formula: see text]. In this paper, we characterize Noetherian and reduced rings over which every (respectively, nonzero) proper ideal is strongly quasi-primary. We also characterize ring over which every strongly quasi primary ideal of [Formula: see text] is prime. Many examples are given to illustrate the obtained results.

Galois criterion for torsion points of Drinfeld modules

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal [Formula: see text] of [Formula: see text], the question essentially asks whether, up to isogeny, a Drinfeld module [Formula: see text] over [Formula: see text] contains a rational [Formula: see text]-torsion point if the reduction of [Formula: see text] at almost all primes of [Formula: see text] contains a rational [Formula: see text]-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2, but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive.

The pascoite family of minerals, including the redefinition of rakovanite

ABSTRACT Mineral species that contain the decavanadate isopolyanion [V10O28]6–, including its protonated and mixed-valence variants, constitute the pascoite family of minerals. Within the pascoite family, the isostructural minerals pascoite and magnesiopascoite form the pascoite group and the isostructural minerals lasalite and ammoniolasalite form the lasalite group. Rakovanite, which was originally assigned the ideal formula Na3[H3V10O28]·15H2O, is redefined with the ideal formula (NH4)3Na3[V10O28]·12H2O.