  For the algebra $\mI_1= K<x, \frac{d}{dx}, \int>$ of polynomial
integro-differential operators over a field $K$ of characteristic zero, a
classification of simple modules is given. It is proved that $\mI_1$ is a left
and right coherent algebra. The {\em Strong Compact-Fredholm Alternative} is
proved for $\mI_1$. The endomorphism algebra of each simple $\mI_1$-module is a
{\em finite dimensional} skew field. In contrast to the first Weyl algebra, the
centralizer of a non-scalar integro-differential operator can be a
noncommutative, non-Noetherian, non-finitely generated algebra which is not a
domain. It is proved that neither left nor right quotient ring of $\mI_1$
exists but there exists the {\em largest left quotient ring} and the {\em
largest right quotient ring} of $\mI_1$, they are not $\mI_1$-isomorphic but
$\mI_1$-{\em anti-isomorphic}. Moreover, the factor ring of the largest right
quotient ring modulo its only proper ideal is isomorphic to the quotient ring
of the first Weyl algebra. An analogue of the Theorem of Stafford (for the Weyl
algebras) is proved for $\mI_1$: each {\em finitely generated} one-sided ideal
of $\mI_1$ is 2-generated.
