Idempotent plethories

Abstract

Let k be a commutative ring with identity. A k-plethory is a commutative k-algebra P together with a comonad structure WP , 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 WP 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 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 Int l (k) = Int(k) ∩ l[e] has the structure, necessarily unique and idempotent, of a k-plethory with unit given by the inclusion k[e] −→ Int l (k). Our results, when applied to the binomial plethory Int(Z), specialize to known results on binomial rings.

Integer-valued polynomials on commutative rings and modules

“…Roughly a decade ago, the author proved a few elementary universal properties of Int(D) for any integral domain D [6]. More recently the author proved some less elementary universal properties of Int(R) for specific classes of rings R [4]. Consider the following conditions on a ring R.…”

“…Date: July 10, 2018. In general, conditions (3)- (5) are equivalent, and they hold if either (1) or (2) holds [4,Theorems 2.9 and 7.11]. Moreover, conditions (4) and (5), when they hold, each provide a universal property for the ring Int(R).…”

“…Date: July 10, 2018. In general, conditions (3)- (5) are equivalent, and they hold if either (1) or (2) holds [4,Theorems 2.9 and 7.11]. Moreover, conditions (4) and (5), when they hold, each provide a universal property for the ring Int(R). It is known, for example, that conditions (1) and (2) hold if R is a Dedekind domain or a UFD, and condition (2) holds if R is a Krull domain or more generally a domain of Krull type [4,Theorem 7.11].…”

“…Moreover, conditions (4) and (5), when they hold, each provide a universal property for the ring Int(R). It is known, for example, that conditions (1) and (2) hold if R is a Dedekind domain or a UFD, and condition (2) holds if R is a Krull domain or more generally a domain of Krull type [4,Theorem 7.11]. Thus, the ring Int(R) can be characterized by the universal properties (4) and (5) for any ring R satisfying any of the conditions (1)- (3), which includes, for example, any domain R of Krull type.…”

“…It is known, for example, that conditions (1) and (2) hold if R is a Dedekind domain or a UFD, and condition (2) holds if R is a Krull domain or more generally a domain of Krull type [4,Theorem 7.11]. Thus, the ring Int(R) can be characterized by the universal properties (4) and (5) for any ring R satisfying any of the conditions (1)- (3), which includes, for example, any domain R of Krull type. The author has conjectured that there exist integral domains D such that Int(D) is not free as a D-module, or more generally such that the equivalent conditions (3)- (5) do not hold for R = D [4,5].…”