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(Definition 2.1 in Bhatt–Scholze.)
Fix a prime $p$. A δ-ring is a pair $(R,δ)$, where $R$ is a commutative ring and $\delta\colon R\to R$ is a map of underlying sets such that $\delta(0)=0$, $\delta(1)=0$,
$\delta(xy)=x^p \delta(y)+y^p \delta(x) + p\delta(x)\delta(y),$and
$\delta(x+y)=\delta(x)+\delta(y)+(x^p+y^p-(x+y)^p)/p.$If $(R,\delta)$ is a δ-ring, then the map $\phi\colon R\to R$ given by $\phi(x)=x^p + p\delta(x)$ is a ring homomorphism that lifts the Frobenius endomorphism on $R/p$.
For $p$-torsionfree rings, the above correspondence between δ-structures and lifts of the Frobenius endomorphism on $R/p$ to $R$ is bijective. This motivates the identities in the definition of a δ-structure.
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