We prove that the algebra In := K x1, .

a b s t r a c tFor the algebra I 1 = K ⟨x, d dx , ⟩ of polynomial integro-differential operators over a field K of characteristic zero, a classification of simple modules is given. It is proved that I 1 is a left and right coherent algebra. The Strong Compact-Fredholm Alternative is proved for I 1 . The endomorphism algebra of each simple I 1 -module is a finite dimensional skew field. In contrast to the first Weyl algebra, the centralizer of a nonscalar 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 I 1 exists but there exists the largest left quotient ring and the largest right quotient ring of I 1 , they are not I 1 -isomorphic but I 1 -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 I 1 : each finitely generated one-sided ideal of I 1 is 2-generated.

Abstract. We present solutions to isomorphism problems for various generalized Weyl algebras, including deformations of type-A Kleinian singularities and the algebras similar to U (sl 2 ) introduced by S. P. Smith. For the former, we generalize results of Dixmier on the first Weyl algebra and the minimal primitive factors of U (sl 2 ) by finding sets of generators for the group of automorphisms.

a b s t r a c tWe study in detail the algebra S n in the title which is an algebra obtained from a polynomial algebra P n in n variables by adding commuting, left (but not two-sided) inverses of the canonical generators of P n . The algebra S n is non-commutative and neither left nor right Noetherian but the set of its ideals satisfies the a.c.c., and the ideals commute. It is proved that the classical Krull dimension of S n is 2n; but the weak and the global dimensions of S n are n. The prime and maximal spectra of S n are found, and the simple S n -modules are classified. It is proved that the algebra S n is central, prime, and catenary. The set I n of idempotent ideals of S n is found explicitly. The set I n is a finite distributive lattice and the number of elements in the set I n is equal to the Dedekind number d n .