Ring of mixed characteristic

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.

Examples

  • The integers Z have characteristic zero, but for any prime number p, Z/(p) is a finite field with p elements and hence has characteristic p.
  • 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.
  • The p-adic integers Zp 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 ZZp. The quotient Zp/pZp is again the finite field of p elements. Zp is an example of a complete discrete valuation ring of mixed characteristic.

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