The authors apply the theory of Noetherian rings to produce a new family of irreducible representations for the Virasoro algebras \(V\) and \(\tilde V\), where \(V\) is the complex Lie algebra with basis \(\{e_ k\}_{k\in\mathbb{Z}}\) with Lie multiplication \([e_ k,e_{\ell}]=(\ell- k)e_{k+\ell}\) and admits a one-dimensional central extension \(\tilde V\) with Lie multiplication \([e_ k,e_{\ell}]=(\ell-k)e_{k+\ell}+{1\over 12} (\ell^ 3-\ell)\delta_{k+\ell,0}c\) for \(c\) a distinguished central element. Let \(L_ 1\) denote the ring \({\mathbb{C}}[x,x^{-1};d/dx]\) of differential operators over the Laurent polynomial ring \({\mathbb{C}}[x,x^{- 1}]\). The authors show that, with one exception, every irreducible \(L_ 1\)-module is an irreducible representation for \(U(V)\), the universal enveloping algebra of \(V\). These algebras are related via a map \(\phi: U(V)\to L_ 1\) extending a Lie homomorphism from \(V\) to \(L_ 1\) defined by the identification of \(e_ i\) with \(x^{i+1}d/dx\).