The purpose of this note is to point out an interesting connection between the structure of a commutative, Noetherian ring of prime characteristic as a (left) module over its ring of diierential operators and various well studied properties such as F-purity, F-regularity, and strong F-regularity. Theorem 2.2 establishes the rst connections between the D-module structure of rings of characterist...

In our work [9], we complete Jans’ classification of TTF-triples [8] by giving a precise description of those two-sided ideals of a ring associated to one-sided split TTF-triples in the corresponding module category.

Let (S,≤) be a strictly totally ordered monoid, R be a commutative ring and M be an R-module. We show the following results: (1) If (S,≤) satisfies the condition that 0 ≤ s for all s ∈ S, then the module [[MS,≤]] of generalized power series is a semi Hopfian [[RS,≤]]-module if and only if M is a semi Hopfian R-module; (2) If (S,≤) is artinian, then the generalized inverse polynomial module [MS,...

The effect of inoculum size and medium volume on methionine production by Bacillus species EZ- 13 ZM-10 was studied. methionine-producing bacteria had already been isolated from within Anambra State University (now Chukwuemeka Odumegwu Ojukwu University), Uli campus State. They were purified identified as EZ-13 using cultural biochemical characteristics. Thereafter, they used to evaluated the i...

light is never being distributed homogeneously and it forms a gradient over biomass. the unidirectional nature of light gives rise to a vertical gradient of light intensity as a function of depth. the maximum depth of the light zone suitable for phytoplankton photosynthesis is designated as the euphotic depth. this study was designed to test the hypothesis of mixing depth of phytoplankton and i...

Let m be a positive integer whose smallest prime divisor is denoted by p, and let Zm denote the cyclic group of residues modulo m. For a set B = {x1, x2, . . . , xm} of m integers satisfying x1 < x2 < · · · < xm, and an integer j satisfying 2 ≤ j ≤ m, define gj(B) = xj − x1. Furthermore, define fj(m, 2) (define fj(m, Zm)) to be the least integer N such that for every coloring ∆ : {1, . . . ,N} ...

Our goal will be to find subsets of Zm that form groups under the operation of multiplication modulo m. By utilizing the isomorphism Zm ∼= Zn ⊕ Zk, we will find multiplicative groups contained in Zn ⊕ Zk and then map these back to Zm. In particular, if m = nk with gcd(n, k) = 1, our objective is to find particular multiplicative subsets of Zn × Zk that are groups and whose first coordinate is a...

Given a ring homomorphism B → A, consider its centralizer R = A, bimodule endomorphism ring S = EndBAB and sub-tensor-square ring T = (A ⊗B A) B . Nonassociative tensoring by the cyclic modules RT or SR leads to an equivalence of categories inverse to the functors of induction of restricted A-modules or restricted coinduction of B-modules in case A |B is separable, H-separable, split or left de...

We discuss the lift of certain D6-antiD6-brane systems to M-theory. These are purely gravitational configurations with a bolt singularity. When reduced along a trivial circle, and for large bolt radius, the bolt is related to a nonsupersymmetric orbifold type of singularity where some closed string tachyons are expected in the twisted sectors. This is a kind of open-closed string duality that r...

The pancreas plays an important role in the homeostasis of zinc (Zn), a nutritionally essential metal. In several previous studies, Zn ions induced inflammatory changes exocrine pancreas; however, little is known about complexes. this study, we microscopically, immunohistochemically, and ultrastructurally examined pancreatic lesions Sprague-Dawley (SD) rats by 4-week repeated oral dose toxicity...