  Let $L$ be an affine Kac-Moody algebra, with central element $c$, and let
$\lambda \in \mathbb C$. We study two-sided ideals in the central quotient
$U_\lambda(L):= U(L)/(c-\lambda)$ of the universal enveloping algebra of $L$,
and prove:
  Theorem 1. If $\lambda \neq 0$ then $U_\lambda(L)$ is simple.
  Theorem 2. The algebra $U_0(L)$ has just-infinite growth, in the sense that
any proper quotient has polynomial growth.
  As an immediate corollary, we show that the annihilator of any nontrivial
integrable highest weight representation of $L$ is centrally generated,
extending a result of Chari for Verma modules.
  We also show that universal enveloping algebras of loop algebras and current
algebras of finite-dimensional simple Lie algebras have just-infinite growth,
and prove similar results to Theorems 1 and 2 for quotients of symmetric
algebras of these Lie algebras by Poisson ideals.
