The notion of a $\lambda$-ring$R$ is closely tied to classical Adams operations$\psi^n \colon R \to R$, modeled by monoidhomomorphisms from the multiplicative monoid $(\mathbb{N}_+, \cdot)$ to the monoid of ring homomorphisms on the underlyingcommutative ring$R$ and their compositions. As $\mathbb{N}_+$ is freely generated by primes $p$, the Adams operations $\psi^p$ play a distinguished role. Classically, they are “Frobenius lifts”, i.e., $\psi^p(x)$ agrees with $x^p$ modulo $p$: the Frobenius map $x \mapsto x^p$ which defines a ring endomorphism on $R/p R$ lifts to the homomorphism $\psi^p\colon R \to R$.

The notion of $\delta$-ring, originally due to André Joyal and developed independently by Alexandru Buium and others, axiomatizes the formal properties that hold for the “arithmetic derivative”

$\delta(x) = \frac{\psi^p(x) - x^p}{p}.$

Definition

Fix a prime number $p$ and a power $q = p^r$ ($r \geq 0$).

Definition

A $\delta$-ring is a commutative ring equipped with a unary operation $\delta: A \to A$ satisfying the following identities: