Derivation on Vinberg rings

A nonassociative ring which contains a well-known associative ring or left symmetric ring also known as Vinberg ring is of great interest. A method to construct Vinberg nonassociative ring is given; Vinberg nonassociative ring is shown as simple; all the derivations of nonassociative simple Vinberg algebra defined are determined; and finally in solid algebra it is shown that if is a nonzero endomorphism of , then is an epimorphism.

Rings with (x, y, z) – (z, y, x) in the right nucleus

Let R be a semiprime nonassociative ring satisfying (x, y, z)–(z, y, x) ∈ Nr then Nl = Nr where Nl and Nr are Lie ideals of R, the set {x ∈ Nr : (R, R, R)x = 0} = {x ∈ Nl : x(R, R, R) = 0} is an ideal of R, and it is contained in the nucleus. Further if [R, R]Nr ⊂ Nr and R is a prime ring with Nr ≠ 0 then R is either associative or commutative.

PRIMITIVITY OF FINITE SEMIFIELDS WITH 64 AND 81 ELEMENTS

A finite semifield D is a finite nonassociative ring with identity such that the set D* = D \{0} is a loop under the product. Wene conjectured in [1] that any finite semifield is either right or left primitive, i.e. D* is the set of right (or left) principal powers of an element in D. In this paper we study the primitivity of finite semifields with 64 and 81 elements.

COORDINATE SETS OF GENERALIZED GALOIS RINGS

A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic pn, for some prime number p, such that its top-factor [Formula: see text] is a semifield. It is well-known that if S is an associative Galois Ring (GR), then it contains a multiplicatively closed subset isomorphic to [Formula: see text], the so-called Teichmüller Coordinate Set (TCS). In this paper we show that the existence of a TCS characterizes GR in the class of all GGR S such that the multiplicative loop [Formula: see text] is right (or left) primitive.

PRIME NONASSOCIATIVE RINGS WITH SKEW DERIVATIONS

Let $R$ be a prime nonassociative ring, $G$ the nucleus of $R$ and $s$, $t$ be automorphisms of $R$. (I) Suppose that $\delta$ is an $s$-derivation of $R$ such that $s\delta=\delta s$ and $\lambda$ is an $t$-derivation of $R$. If $\lambda\delta^n=0$ and $\delta^n(R)\subset G$, where $n$ is a fixed positive integer, then $\lambda=0$ or $\delta^{3n-1}=0$. (II) Assume that $\delta$ and $\lambda$ are derivations of $R$. If there exists a fixed positive integer $n$ such that $\lambda^n\delta=0$, and $\delta(R)\subset G$ or $\lambda^n(R)\subset G$, then $\delta^2= 0$ or $\lambda^{6n-4}= 0$.

RINGS WITH A JORDAN DERIVATION WHOSE IMAGE IS CONTAINED IN THE NUCLEI OR COMMUTATIVE CENTER

Let $R$ be a nonassociative ring, $N$, $L$ and $G$ the left nucleus, right nucleus and nucleus respectively. It is shown that if $R$ is a prime ring with a Jordan derivation d such that $d(R) \subseteq G$ and $(d^2 R), R) \subseteq N$ or $(d^2(R), R) \subseteq L$ then either $R$ is associative or $2d^2 =0$. Moreover. if $(d(R), R) =0$ then either $R$ is associative and commutative or $2d =0$. We also prove that if $R$ is a prime ring with a derivation $d$ and there exists a fixed positive integer $n$ such that $d^n(R) \subseteq G$ and $(d^n(R), R) =0$ then $R$ is associative and $d^n =0$, or $R$ is associative and commutative, or $d^{2n} = (\frac{(2n)!}{n!})d^n = 0$. This partially generalize the results of [3]. We also obtain some results on prime rings with a derivation satisfying other hypotheses.

RINGSWITH (R,R,R)AND [R,(R,R,R)] IN THE LEFT NUCLEUS

Let $R$ be a nonassociative ring, and $N = (R,R,R) + [R,(R,R,R)]$. We show that $W = \{w\in N | Rw +wR +R(wR) \subset N\}$ is a two-sided ideal of $R$. If for some $r\in R$, any one of the sets $(r, R, R)$, $(R,r, R)$ or $(R, R,r)$ is contained in $W$, then the other two sets are contained in $W$ also. If the associators are assumed to be contained in either the left, the middle, or the right nucleus, and $I$ is the ideal generated by all associators, then $I^2 \subset W$. If $N$ is assumed to be contained in the left or the right nucleus, then $W^2 = 0$. We conclude that if $R$ is semiprime and $N$ is contained in the left or the right nucleus, then $R$ is associative. We assume characteristic not 2.

NONASSOCIATIVE RINGS WITH A SPECIAL DERIVATION

Let $R$ be a nonassociative ring, $N$, $L$ and $G$ the left nucleus, right nucleus and nucleus respectively. It is shown that if $R$ is a prime ring with a derivafion $d$ such that $ax+ d(x) \in G$ where $a \in Z$, the ring of rational integers, or $a \in G$ with $(ad)(x) = ad(x) = d(ax)$ and $ax = xa$ for all $x$ in $R$ then either $R$ is associative or $ad+ d^2 = 2d(R)^2 = 0$. This result is also valid under the weaker hypothesis $ax+ d(x) \in N \cap L$ for all $x$ in $R$ for the simple ring case, and we obtain that either $R$ is associative or $((ad+ d^2)(R))^2 =0$ for the prime ring case.

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.