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.

Examples

 * The integers $$\mathbb{Z}$$ have characteristic zero, but for any prime number $$p$$, $$\mathbb{F}_p=\mathbb{Z}/p\mathbb{Z}$$ 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 Z(p)&thinsp;/I 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 $$\mathcal{O}_K$$ of integers of a number field $$K$$, then the localization of $$\mathcal{O}_K$$ at $$P$$ is likewise of mixed characteristic.
 * The p-adic integers Zp 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 Z &rarr; Zp. The quotient Zp/pZp is again the finite field of p elements. Zp 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.