Derivations and right ideals of algebras

Proceedings of the American Mathematical Society, 1993

Let R be a semiprime A^-algebra with unity, d a nonzero derivation of R , and f(xx, ... , xi) a monic multilinear polynomial over K such that d(f(ax, ... , at)) / 0 for some ax, ... , at £ R. It is shown that if for every rx, ... ,rt in R either d(f(rx, ... , rt)) = 0 or d(f(rx, ... , rt)) is invertible in R , then R is either a division ring D or M2(D), the ring of 2 x 2 matrices over D , unless f(xx, ... , xt) is a central polynomial for R. Moreover, if R = M2(D), where 2R ^ 0 and f(xx, ... , xt) is not a central polynomial for D , then d is an inner derivation of R .

Kyoto Journal of Mathematics

Proceedings of the 15th WSEAS international …, 2010

We study properties of a differentially simple commutative ring R with respect to a set D of derivations of R. Among the others we investigate the relation between the D-simplicity of R and that of the local ring R P with respect to a prime ideal P of R and we prove a criterion about the D-simplicity of R in case where R is a 1-dimensional (Krull dimension) finitely generated algebra over a field of characteristic zero and D is a singleton set. The above criterion was quoted without proof in an earlier paper of the author.

Differential Galois Theory, 2002

Communications in Algebra, 2001

2006

Let $A$ be an integral $k$-algebra of finite type over a field $k$ of characteristic zero. Let ${\\cal{F}}$ be a family of $k$-derivations on $A$ and $M_{\\cal{F}}$ the $A$-module spanned by ${\\cal{F}}$. In this paper, we generalize a result due to A. Nowicki and construct an element $\\partial$ of $M_{\\cal{F}}$ such that $\\ker \\partial=\\cap_{d\\in {\\cal{F}}} \\ker d$. Such a derivation is called ${\\cal{F}}$-minimal. Then we establish a density theorem for ${\\cal{F}}$-minimal derivations in $M_{\\cal{F}}$.

Kyoto Journal of Mathematics

Journal of Linear and Topological Algebra, 2017

The main purpose of this article is to offer some characterizations of $delta$-double derivations on rings and algebras. To reach this goal, we prove the following theorem:Let $n > 1$ be an integer and let $mathcal{R}$ be an $n!$-torsion free ring with the identity element $1$. Suppose that there exist two additive mappings $d,delta:Rto R$ such that $$d(x^n) =Sigma^n_{j=1} x^{n-j}d(x)x^{j-1}+Sigma^{n-2}_{k=0} Sigma^{n-2-k}_{i=0} x^kdelta(x)x^idelta(x)x^{n-2-k-i}$$ is ful filled for all $xin mathcal{R}$. If $delta(1) = 0$, then $d$ is a Jordan $delta$-double derivation. In particular, if $mathcal{R}$ is a semiprime algebra and further, $delta^2(x^2) = delta^2(x)x + xdelta^2(x) + 2(delta(x))^2$ holds for all $xin mathcal{R}$, then $d-frac{1}{2}delta^2$ is an ordinary derivation on $mathcal{R}$.

2012

Let R be a 2-torsion free ring and let U be a square closed Lie ideal of R. Suppose that α, β are automorphisms of R. An additive mapping δ : R −→ R is said to be a Jordan left (α, β)-derivation of R if δ(x 2) = α(x)δ(x) + β(x)δ(x) holds for all x ∈ R. In this paper it is established that if R admits an additive mapping G : R −→ R satisfying G(u 2) = α(u)G(u) + α(u)δ(u) for all u ∈ U and a Jordan left (α, α)-derivation δ; and U has a commutator which is not a left zero divisor, then G(uv) = α(u)G(v) + α(v)δ(u) for all u, v ∈ U. Finally, in the case of prime ring R it is proved that if G : R −→ R is an additive mapping satisfying G(xy) = α(x)G(y) + β(y)δ(x) for all x, y ∈ R and a left (α, β)-derivation δ of R such that G also acts as a homomorphism or as an anti-homomorphism on a nonzero ideal I of R, then either R is com-mutative or δ = 0 on R.

Bulletin of the iranian mathematical society

Let R be a 2-torsion free ring and let U be a square closed Lie ideal of R. Suppose that α, β are automorphisms of R. An additive mapping δ : R −→ R is said to be a Jordan left (α, β)derivation of R if δ(x 2) = α(x)δ(x) + β(x)δ(x) holds for all x ∈ R. In this paper it is established that if R admits an additive mapping G : R −→ R satisfying G(u 2) = α(u)G(u) + α(u)δ(u) for all u ∈ U and a Jordan left (α, α)-derivation δ; and U has a commutator which is not a left zero divisor, then G(uv) = α(u)G(v) + α(v)δ(u) for all u, v ∈ U. Finally, in the case of prime ring R it is proved that if G : R −→ R is an additive mapping satisfying G(xy) = α(x)G(y) + β(y)δ(x) for all x, y ∈ R and a left (α, β)derivation δ of R such that G also acts as a homomorphism or as an anti-homomorphism on a nonzero ideal I of R, then either R is commutative or δ = 0 on R.