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.{{citation

| last1 = Bergman | first1 = George M. | author1-link = George Bergman

| last2 = Hausknecht | first2 = Adam O.

| doi = 10.1090/surv/045

| isbn = 0-8218-0495-2

| mr = 1387111

| page = 336

| publisher = American Mathematical Society, Providence, RI

| series = Mathematical Surveys and Monographs

| title = Co-groups and co-rings in categories of associative rings

| url = https://books.google.com/books?id=s6NnkQs3JBMC&pg=PA336

| volume = 45

| year = 1996}}.

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 {{nowrap|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 {{nowrap|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.