Ring of mixed characteristic

In commutative algebra, a ring of mixed characteristic is a commutative ring having characteristic zero and having an ideal such that has positive 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 (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 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.

References