## Aspects of affine algebras |

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additivity principle affine associated graded algebra associative algebra commutative integral domain contain Corollary defined Definition denote direct sum enveloping superalgebra exists a non-zero field filtered algebra finite codimension finite composition length finite dimensional Lie Gr(A Gr(M graded algebra hence Hilbert's Nullstellensatz homogeneous basis homomorphic image ideal Q implies induction integral domain intersection isomorphic Jacobson radical Jacobson ring k-algebra satisfying GF L.W. Small left R-module left right Lemma Lie algebras Lie superalgebra Math Mathematics module nilpotent Noetherian almost PI Noetherian ring non-zero element normalizing extension obtain poly-(finite/almost normalizing prime ideal primitive Proof proper factor Proposition prove R-submodules regular elements right ideal right Noetherian ring extension ring of quotients ring theory satisfies generic flatness satisfies GF/K satisfies the Nullstellensatz section 2.2 semiprime semisimple simple R-module submodule subspace superbracket Theorem two-sided ideal U(LQ universal enveloping algebra vector space Z2-graded zero divisors