208 NOTICES OF THE AMS VOLUME 51, NUMBER 2 As far as I know, the notion of a quasi-morphism does not have much to do with category theory. This very natural idea underlies several interesting recent developments at the crossroads of algebra, topology, geometry, and dynamics. Other, perhaps better, names currently in use for the same concept are quasi-homomorphism and pseudo-character. Let G be ...

A new relation between Prym varieties of arbitrary morphisms of algebraic curves and integrable systems is discovered. The action of maximal commutative subalgebras of the formal loop algebra of GLn defined on certain infinite-dimensional Grassmannians is studied. It is proved that every finite-dimensional orbit of the action of traceless elements of these commutative Lie algebras is isomorphic...

Exercise 10.3.2. Let R be a commutative ring with identity. For all positive integers n and m, R ∼= R if and only if n = m. Proof. Let φ : R → R be an isomorphism of R-modules and let I E R be a maximal ideal. Then the map φ̄ : R → R/IR given by φ̄(α) = φ(α) is a morphism of R-modules. Moreover ker φ̄ = {α ∈ R | φ̄(α) = 0} = {α ∈ R | φ(α) ∈ IR} = φ−1(IRm) = IR. Therefore by the first isomorphism th...