commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...

, a ring of mixed characteristic is a commutative ring

R

having characteristic zero and having an

ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...

I

such that

R/I

has positive characteristic..

Examples

* The

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

\mathbb

have characteristic zero, but for any

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

p

\mathbb_p=\mathbb/p\mathbb

is a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...

with

p

elements and hence has characteristic

p

. * The

ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...

of any

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

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 In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...

''p''Z_(''p''), and the

quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

Z_(''p'')/''p''Z_(''p'') is a finite field with ''p'' elements. In contrast to the previous example, the only possible characteristics for rings of the form are zero (when ''I'' is the

zero ideal In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive ident ...

) 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

\mathcal_K

of integers of a number field

K

, then the

localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is af ...

\mathcal_K

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 An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...

of the prime number ''p'' under the canonical map . The quotient Z_''p''/''p''Z_''p'' is again the finite field of ''p'' elements. Z_''p'' is an example of a

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies ...

discrete valuation ring In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R' ...

of mixed characteristic. * The integers, the

of any number field, and any localization or completion of one of these rings is a characteristic zero

Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessari ...

References

Commutative algebra {{algebra-stub