Ring of mixed characteristic explained

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/pZ} 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 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 l{O} _{K} of integers of a number field K , then the localization of l{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 . 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.

Notes and References

1. .