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.

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 ${\displaystyle p}$ and localize the integers at the prime ideal ${\displaystyle (p)}$. The resulting ring ${\displaystyle \mathbb {Z} _{p}}$ has characteristic zero. It has a unique maximal ideal ${\displaystyle p\mathbb {Z} _{(p)}}$, and the quotient ${\displaystyle \mathbb {Z} _{(p)}/p\mathbb {Z} _{(p)}}$ is a finite field with ${\displaystyle p}$ elements. In contrast to the previous example, the only possible characteristics for rings of the form ${\displaystyle \mathbb {Z} _{(p)}/I}$ are zero (when ${\displaystyle I}$ is the zero ideal) and powers of ${\displaystyle p}$ (when ${\displaystyle I}$ 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 ${\displaystyle p}$-adic integers ${\displaystyle \mathbb {Z} _{p}}$ for any prime ${\displaystyle p}$ are a ring of characteristic zero. However, they have an ideal generated by the image of the prime number ${\displaystyle p}$ under the canonical map ${\displaystyle \mathbb {Z} \rightarrow \mathbb {Z} _{p}}$. The quotient ${\displaystyle \mathbb {Z} _{p}/p\mathbb {Z} _{p}}$ is again the finite field of ${\displaystyle p}$ elements. ${\displaystyle \mathbb {Z} _{p}}$ is an example of a complete discrete valuation ring of mixed characteristic.

References

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