Simplicity theorems for Lie algebras of derivations of commutative rings (Q919447)

Let R be an associative commutative ring with 1 and Der R be the Lie ring of all its derivations. The author defines certain Lie subalgebras of Der R generalizing the Lie algebras of Cartan type and proves simplicity theorems for these algebras. For example, let R admit a Lie ring structure [, ] which is connected with the original multiplication in R by the rule \[ [x,yz]-[x,y]z-y[x,z]+[x,1]yz=0. \] (The author calls such operation [, ] the Poisson-Jacobi bracket.) Set \(\partial (x)=[1,x]\), \(\{x,y\}=[x,y]-x\partial (y)+y\partial (x)\); \(\rho (x)(y)=x\partial (y)+\{x,y\}\); then \(\rho\) (x)\(\in Der R\) and \(\rho ([x,y])=[\rho (x),\rho (y)]\) which means that the mapping \(\rho\) : \(R\to Der R\) is a Lie ring homomorphism. The Lie ring \(L=\rho (R)\) generalizes the Lie algebras of Hamiltonian and contact types. It is proved, in particular that if \(L\neq 0\), \(6R=R\) and R is L-simple, then the Lie ring [L,L] is simple.