Ring of mixed characteristic

In commutative algebra, a ring of mixed characteristic is a commutative ring $R$ having characteristic zero and having an ideal $I$ such that $R / I$ has positive characteristic.^[1]

Examples

The integers $ℤ$ have characteristic zero, but for any prime number $p$ , $𝔽_{p} = ℤ / p ℤ$ is a finite field with $p$ elements and hence has characteristic $p$ .
The ring of integers of any number field is of mixed characteristic
Fix a prime p and localize the integers at the prime ideal (p). The resulting ring Z_(p) has characteristic zero. It has a unique maximal ideal pZ_(p), and the quotient Z_(p)/pZ_(p) is a finite field with p elements. In contrast to the previous example, the only possible characteristics for rings of the form Template:Nowrap are zero (when I is the zero ideal) and powers of p (when I is any other non-unit ideal); it is not possible to have a quotient of any other characteristic.
If $P$ is a non-zero prime ideal of the ring $𝒪_{K}$ of integers of a number field $K$ , then the localization of $𝒪_{K}$ at $P$ is likewise of mixed characteristic.
The p-adic integers Z_p for any prime p are a ring of characteristic zero. However, they have an ideal generated by the image of the prime number p under the canonical map Template:Nowrap. The quotient Z_p/pZ_p is again the finite field of p elements. Z_p is an example of a complete discrete valuation ring of mixed characteristic.
The integers, the ring of integers of any number field, and any localization or completion of one of these rings is a characteristic zero Dedekind domain.