Simple algebras of Weyl type

Over a field $F$ of any characteristic, for a commutative associative algebra $A$ with an identity element and for the polynomial algebra $F[D]$ of a commutative derivation subalgebra $D$ of $A$, the associative and the Lie algebras of Weyl type on the same vector space $A[D]=A\otimes F[D]$ are defined. It is proved that $A[D]$, as a Lie algebra (modular its center) or as an associative algebra, is simple if and only if $A$ is $D$-simple and $A[D]$ acts faithfully on $A$. Thus a lot of simple algebras are obtained.

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