  We consider a non-Hermitian random matrix $A$ whose distribution is invariant
under the left and right actions of the unitary group. The so-called Single
Ring Theorem, proved by Guionnet, Krishnapur and Zeitouni, states that the
empirical eigenvalue distribution of $A$ converges to a limit measure supported
by a ring $S$. In this text, we establish the convergence in distribution of
random variables of the type $Tr (f(A)M)$ where $f$ is analytic on $S$ and the
Frobenius norm of $M$ has order $\sqrt{N}$. As corollaries, we obtain central
limit theorems for linear spectral statistics of $A$ (for analytic test
functions) and for finite rank projections of $f(A)$ (like matrix entries). As
an application, we locate outliers in multiplicative perturbations of $A$.
