  We construct homomorphisms from the universal enveloping algebra of the
positive (part of the) Witt algebra to several different Artin-Schelter regular
algebras, and determine their kernels and images. As a result, we produce
elementary proofs that the universal enveloping algebras of the Virasoro
algebra, the Witt algebra, and the positive Witt algebra are neither left nor
right noetherian.
