Maximal ideal

A proper ideal

m

of a commutative ring

R

is maximal when no proper ideal lies strictly between

m

and

R

. Equivalently,

R / m

is a field.

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Residue field ( $κ (m) = R / m$ )

The residue field at a maximal ideal

m

is the quotient field

κ (m) = R / m

. At a prime ideal

p

, the residue field is the fraction field of

R / p

.

Residue-field extension

A local homomorphism of local rings

(R, m) \to (S, n)

induces the residue-field extension

R / m \to S / n

. For an extension of non-Archimedean local fields, its degree is the residue-field degree.

Residue characteristic

The residue characteristic of a local ring is the characteristic of a ring of its residue field. A finite extension of

Q_{p}

has residue characteristic

p

.

Maximal spectrum ( $MaxSpec R$ )

The maximal spectrum

MaxSpec R

is the set of maximal ideals of a commutative ring

R

.

Maximal ideal quotient criterion

m

of a commutative unital ring

R

is maximal exactly when

R / m

is a field.

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Maximal ideal by

Wikipedia Bot  1

In abstract algebra, a **maximal ideal** is a specific type of ideal within a ring. To define it, let's first recall some basic concepts related to rings and ideals: 1. **Ring**: A set equipped with two binary operations, typically called addition and multiplication, satisfying certain properties (like associativity, distributivity, etc.). 2. **Ideal**: A subset of a ring that absorbs multiplication by elements from the ring and is closed under addition.

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