Ring of mixed characteristic - Alchetron, the free social encyclopedia

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 , F p = Z / p Z is a finite field with p elements and hence has characteristic 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 is a non-zero prime ideal of the ring O K of integers of a number field K then the localization of O K at 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.

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

Ring of mixed characteristic Wikipedia

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