The algebra of integro-differential operators on an affine line and its modules

Abstract: 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.