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 $\mathbb{Z}$ 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) / 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 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 Z → Z_p. 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.

References

1. Bergman, George M.; Hausknecht, Adam O. (1996), Co-groups and co-rings in categories of associative rings, Mathematical Surveys and Monographs 45, American Mathematical Society, Providence, RI, p. 336, doi:10.1090/surv/045, ISBN 0-8218-0495-2, MR 1387111.