Idempotent plethories

Let k be a commutative ring with identity. A k-plethory is a commutative k-algebra P   together with a comonad structure WPWP, called the P-Witt ring functor, on the covariant functor that it represents. We say that a k-plethory P is idempotent   if the comonad WPWP is idempotent, or equivalently if the map from the trivial k  -plethory k[e]k[e] to P is a k-plethory epimorphism. We prove several results on idempotent plethories. We also study the k  -plethories contained in K[e]K[e], where K is the total quotient ring of k  , which are necessarily idempotent and contained in Int(k)={f∈K[e]:f(k)⊆k}Int(k)={f∈K[e]:f(k)⊆k}. For example, for any ring l between k and K we find necessary and sufficient conditions—all of which hold if k   is a integral domain of Krull type—so that the ring Intl(k)=Int(k)∩l[e]Intl(k)=Int(k)∩l[e] has the structure, necessarily unique and idempotent, of a k  -plethory with unit given by the inclusion k[e]⟶Intl(k)k[e]⟶Intl(k). Our results, when applied to the binomial plethory Int(Z)Int(Z), specialize to known results on binomial rings.