On symplectic semifield spreads of PG(5,q 2), q odd

We prove that there exist exactly three non-equivalent symplectic semifield spreads of{\operatorname{PG}(5,q^{2})}, for{q^{2}>2\cdot 3^{8}}odd, whose associated semifield has center containing{\mathbb{F}_{q}}. Equivalently, we classify, up to isotopy, commutative semifields of order{q^{6}}, for{q^{2}>2\cdot 3^{8}}odd, with middle nucleus containing{\mathbb{F}_{q^{2}}}and center containing{\mathbb{F}_{q}}.

Remaining at the Margin and in the Center

Background: Looking up the terms center and margin, synonyms found for center are “the middle, nucleus, heart, core, hub” and synonyms for margin are “the border, edge, boundary, fringe, periphery.” These terms and their synonyms prompt me to ask questions about the concepts of “margin” and “center” as related to higher education. Questions such as: What are the challenges and benefits of being at the margins in the academy? What are the risks and benefits of moving to center? Can faculty of color move to center and continue to remain forces for change in the academy? Are there ways in which one can remain at the margin and in the center simultaneously?

The ideals in (-1,1) rings

A (-1, 1) ring \(R\) contains a maximal ideal \(I_{3}\) in the nucleus \(N\). The set of elements \(n\) in the nucleus which annihilates the associators in (-1, 1) ring \(R\), \(n(x, y, z) = 0\) and \((x, y, z)n = 0\) for all \(x, y, z \in R\) form the ideal \(I_{3}\) of \(R\). Let \(I\) be a right ideal of a 2-torsion free (-1, 1) ring \(R\) with commutators in the middle nucleus. If \(I\) is maximal and nil, then \(I\) is a two sided ideal. Also if \(I\) is minimal then it is either a two-sided ideal, or the ideal it generates is contained in the middle nucleus of \(R\) and the radical of \(R\) is contained in \(P\) for any primitive ideal \(p\) of \(R\).

NUCLEAR SEMIDIRECT PRODUCT OF COMMUTATIVE AUTOMORPHIC LOOPS

Automorphic loops are loops where all inner mappings are automorphisms. We study when a semidirect product of two abelian groups yields a commutative automorphic loop such that the normal subgroup lies in the middle nucleus. With this description at hand we give some examples of such semidirect products.

Semifields in Class ${\cal F}_4^{(a)}$

The semifields of order $q^6$ which are two-dimensional over their left nucleus and six-dimensional over their center have been geometrically partitioned into six classes by using the associated linear sets in $PG(3,q^3)$. One of these classes has been partitioned further (again geometrically) into three subclasses. In this paper algebraic curves are used to construct two infinite families of odd order semifields belonging to one of these subclasses, the first such families shown to exist in this subclass. Moreover, using similar techniques it is shown that these are the only semifields in this subclass which have the right or middle nucleus which is two-dimensional over the center. This work is a non-trivial step towards the classification of all semifields that are six-dimensional over their center and two-dimensional over their left nucleus.

WEAKLY NON-ASSOCIATIVE ALGEBRAS AND THE KADOMTSEV–PETVIASHVILI HIERARCHY

On any ‘weakly non-associative’ algebra there is a universal family of compatible ordinary differential equations (provided that differentiability with respect to parameters can be defined), any solution of which yields a solution of the Kadomtsev–Petviashvili (KP) hierarchy with dependent variable in an associative sub-algebra, the middle nucleus.

RINGS WITH A DERIVATION WHOSE IMAGE IS ZERO ON THE ASSOCIATORS

Let $R$ be a nonassociative ring, $N$, $M$, $L$ and $G$ the left nucleus,middle nucleus, right nucleus and nucleus respectively. Assume that $R$ is a ring with a derivation $d$ such that $d((R, R, R)) = 0$. It is shown that if $R$ is a simple ring then either $R$ is associative or $d(N \cap L) = 0$; and if $R$ is a prime ring satisfying $Rd(G) \subseteq N$ and $d(G)R \subseteq L$, or $d(G)R +Rd(G) \subseteq M$ then either $R$ is associative or $d(G) =0$. These partially extend our previous results.

RINGS WITH A DERIVATION WHOSE IMAGE IS CONTAINED IN THE NUCLEI

Let $R$ be a nonassociative ring, $N$, $M$, $L$ and $G$ the left nucleus, middle nucleus, right nucleus and nucleus respectively. Suh [4] proved that if $R$ is a prime ring with a derivation dsuch that $d(R) \subseteq G$ then either $R$ is associative or $d^3 =0$. We improve this result by concluding that either $R$ is associative or $d^2 =2d =0$ under the weaker hypothesis $d(R)\subseteq N$\cap M$ or $d(R)\subseteq N\cap M$ or $d(R)\subseteq M\cap L$. Using our result, we obtain the theorems of Posner [3] and Yen [11] for the prime nonassociative rings. In our recent papers we partially generalize the above main result.

RINGS WITH ASSOCIATORS IN THE LEFT AND MIDDLE NUCLEUS

Let $R$ be a nonassociative ring, $N$, $M$ and $L$ the left, middle and right nucleus respectively. It is shown that if $R$ a semipnme ring satisfying $(R,R,R) \subset N\cap M$ (resp. $(R,R,R) \subset M\cap L$), then $L\subset M\subset N$(resp. $N\subset M\subset L$); moreover, $R$ is associative if $((R,R,M),(R,R,R)) = 0$ (resp. $((M,R,R),(R,R,R)) = 0)$ or $(M,R) \subset M$; and the Abelian group $(R,+ )$ has no elements of order 2. We also prove that if $R$ is a simple ring satisfying char $R \neq 2$, and $(R, R, R) \subset N \cap M$ or $(R, R, R) \subset M \cap L$ then $R$ is associative.