
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 be a commutative ring with identity. A {\it -plethory} is a commutative -algebra together with a comonad structure , called the {\it -Witt ring} functor, on the covariant functor that it represents. We say that a -plethory is {\it idempotent} if the command is idempotent, or equivalently if the map from the trivial -plethory to is a -plethory epimorphism. We prove several results on idempotent plethories. We also study the -plethories contained in , where is the total quotient ring of , which are necessarily idempotent and contained in . For example, for any ring between and we find necessary and sufficient conditions---all of which hold if is a integral domain of Krull type---so that the ring $\operatorname{Int}_l(k) = \operatorname{Int}(k)…
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Taxonomy
TopicsArchaeology and Historical Studies
