Ring of mixed characteristic

Examples

• The integers have characteristic zero, but for any prime number, is a finite field with elements and hence has characteristic.
• The ring of integers of any number field is of mixed characteristic
• Fix a prime p and localize the integers at the prime ideal. The resulting ring Z has characteristic zero. It has a unique maximal ideal pZ, and the quotient Z'/pZ is a finite field with p'' elements. In contrast to the previous example, the only possible characteristics for rings of the form are zero and powers of p ; it is not possible to have a quotient of any other characteristic.
• If is a non-zero prime ideal of the ring of integers of a number field, then the localization of at 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.