This paper presents two extensions of the second order polymorphiclambda calculus, system F, with monotone (co)inductive types supporting(co)iteration, primitive (co)recursion and inversion principles asprimitives. One extension is inspired by the usual categoricalapproach to programming by means of initial algebras and finalcoalgebras; whereas the other models dialgebras, and can be seen as an extension of Hagino'scategorical lambda calculus within the framework of parametricpolymorphism. The systems are presented in Curry-style, and are proven to be terminating andtype-preserving. Moreovertheir expressiveness is shown by means of several programmingexamples, going from usual data types to lazy codata types such as streamsor infinite trees.

Herstein showed that the conjugacy class of a non-central element in the multiplicative group of a division ring is infinite. We prove similar results for units in algebras and orders and give applications to group rings.

AMS 2000 Mathematics subject classification: Primary 16U60. Secondary 16H05; 16S34; 20F24; 20C05

All rings have an identity, all homomorphisms map identities to identities, all homomorphisms on algebras over fields are algebra homomorphisms. A ring R is a quotient-embeddable ring (a QE-ring) if for any proper ideal a of R there is an endomorphism of R whose kernel is the ideal a. A QE-ring U is a receptor of R if for any proper ideal a of R there is a homomorphism from R to U whose kernel is the ideal a.

Theorem. A ring R has a receptor if and only if it is a K-algebra over some field K contained in the center of R. If R is a commutative K-algebra of this type, then it has a commutative receptor.