Ring of mixed characteristic

In commutative algebra, a ring of mixed characteristic is a commutative ring [math]\displaystyle{ R }[/math] having characteristic zero and having an ideal [math]\displaystyle{ I }[/math] such that [math]\displaystyle{ R/I }[/math] has positive characteristic.

Examples

• The integers [math]\displaystyle{ \mathbb{Z} }[/math] have characteristic zero, but for any prime number [math]\displaystyle{ p }[/math], [math]\displaystyle{ \mathbb{F}_p=\mathbb{Z}/p\mathbb{Z} }[/math] is a finite field with [math]\displaystyle{ p }[/math] elements and hence has characteristic [math]\displaystyle{ p }[/math].
• 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 [math]\displaystyle{ P }[/math] is a non-zero prime ideal of the ring [math]\displaystyle{ \mathcal{O}_K }[/math] of integers of a number field [math]\displaystyle{ K }[/math], then the localization of [math]\displaystyle{ \mathcal{O}_K }[/math] at [math]\displaystyle{ P }[/math] 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.