Idempotent plethories

TL;DR

This paper introduces and studies idempotent $k$-plethories, exploring their properties, structure, and conditions under which certain rings of integer-valued polynomials form such plethories, extending known results on binomial rings.

Contribution

It characterizes idempotent $k$-plethories, especially those within $K[e]$, and establishes conditions for rings of integer-valued polynomials to have a unique idempotent plethory structure.

Findings

Idempotent plethories are characterized by specific algebraic conditions.

Rings of integer-valued polynomials can form unique idempotent plethories under certain conditions.

Results generalize known properties of binomial rings to broader algebraic contexts.

Abstract

Let $k$ be a commutative ring with identity. A {\it $k$-plethory} is a commutative $k$-algebra $P$ together with a comonad structure $W_P$, called the {\it $P$-Witt ring} functor, on the covariant functor that it represents. We say that a $k$-plethory $P$ is {\it idempotent} if the command $W_P$ is idempotent, or equivalently if the map from the trivial $k$-plethory $k[e]$ to $P$ is a $k$-plethory epimorphism. We prove several results on idempotent plethories. We also study the $k$-plethories contained in $K[e]$, where $K$ is the total quotient ring of $k$, which are necessarily idempotent and contained in $\operatorname{Int}(k) = \{f \in K[e]: f(k) \subseteq k\}$. 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 $\operatorname{Int}_l(k) = \operatorname{Int}(k) \cap l[e]$ has the structure, necessarily unique and idempotent, of a $k$-plethory with unit given by the inclusion $k[e] \longrightarrow \operatorname{Int}_l(k)$. Our results, when applied to the binomial plethory $\operatorname{Int}({\mathbb Z})$, specialize to known results on binomial rings.