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Ring of mixed characteristic

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

Examples

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

References

  1. Bergman, George M.; Hausknecht, Adam O. (1996), Co-groups and co-rings in categories of associative rings, Mathematical Surveys and Monographs, vol. 45, American Mathematical Society, Providence, RI, p. 336, doi:10.1090/surv/045, ISBN 0-8218-0495-2, MR 1387111.


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