Idempotents in Intersection of the Kernel and the Image of Locally Finite Derivations and $\mathcal E$-derivations

DOI10.1007/S40879-017-0209-6arXiv1701.05993MaRDI QIDQ6282200

Wenhua Zhao

Publication date: 21 January 2017

Abstract: Let $K$ be a field of characteristic zero, $mathcalA$ a $K$-algebra and $delta$ a $K$-derivation of $mathcalA$ or $K$-$mathcalE$-derivation of $mathcalA$ (i.e., $delta=operatorname{Id}_A-phi$ for some $K$-algebra endomorphism $phi$ of $mathcalA$). Motivated by the Idempotent conjecture proposed in [Z4], we first show that for every idempotent $e$ lying in both the kernel ${mathcalA}^{d}elta$ and the image $operatornameImdelta!:=delta(mathcalA)$ of $delta$, the principal ideal $(e)subseteqoperatornameImdelta$ if $delta$ is a locally finite $K$-derivation or a locally nilpotent $K$-$mathcalE$-derivation of $mathcalA$; and $emathcalA,mathcalAesubseteqoperatornameImdelta$ if $delta$ is a locally finite $K$-$mathcalE$-derivation of $mathcalA$. Consequently, the Idempotent conjecture holds for all locally finite $K$-derivations and all locally nilpotent $K$-$mathcalE$-derivations of $mathcalA$. We then show that $1_{mathcalA}inoperatornameImdelta$, (if and) only if $delta$ is surjective, which generalizes the same result [GN, W] for locally nilpotent $K$-derivations of commutative $K$-algebras to locally finite $K$-derivations and $K$-$mathcalE$-derivations $delta$ of all $K$-algebras $mathcalA$.

Mathematics Subject Classification ID

Commutators, derivations, elementary operators, etc. (47B47) Derivations, actions of Lie algebras (16W25) Automorphisms and endomorphisms of algebraic structures (08A35) Modules, bimodules and ideals in associative algebras (16D99)

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