Ring of mixed characteristic

In commutative algebra, a ring of mixed characteristic is a commutative ring ${\displaystyle R}$ having characteristic zero and having an ideal ${\displaystyle I}$ such that ${\displaystyle R/I}$ has positive characteristic. ^[1]

Examples

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

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 .