Commutative algebra Definition
Maximal ideal
Builds on Ideal
Definition
Let be a ring. An ideal of is maximal if it has these two properties.
- .
- For every ideal of with , either or .
Remarks
The definition is the same for left, right and two-sided ideals, with ideals of the same type in condition 2. In , the ideal with is maximal if and only if is a prime number. In a commutative ring , an ideal is maximal if and only if is a field. The zero ideal of a field is its only maximal ideal.
Reading path
To understand this page from the ground up, read these 4 pages in order. The foundations come first.
Show the path as a graph
- no evidence level
- Group concept
- Ring definition
- Ideal definition
- Maximal ideal definition
Where this leads
36 pages build on this one.
- Uniqueness of finite fields of a given order
- Existence of splitting fields
- Uniqueness of the algebraic closure
- Existence of maximal ideals
- Galois field
- Maximal ideals are prime
- Uniqueness of splitting fields
- In a PID a nonzero element is irreducible iff it generates a maximal ideal
- K[x]/(f) is a field iff f is irreducible
- Impossibility of doubling the cube
- Existence of an algebraic closure
- Impossibility of trisecting the angle
- Adjoining a root of a polynomial
- Degree of a simple extension is the degree of the minimal polynomial
- The degree of a constructible number is a power of two
- Existence of finite fields of every prime power order
- Every polynomial has a root in some extension field
- R/M is a field iff M is maximal
- Impossibility of squaring the circle
- Gauss–Wantzel theorem
- K(a) is isomorphic to K[x] modulo the minimal polynomial
- Model theory of algebraically closed fields
- Jacobson radical
- Weil conjectures
- Jacobson's conjecture
- Akizuki–Hopkins–Levitzki theorem
- Steinitz's theorem on algebraically closed fields
- Standard conjectures on algebraic cycles
- Structure theorem for Artinian rings
- Levitzki's theorem
- Lenagan's theorem
- Categorical theory
- Hopkins–Levitzki theorem
- Impossibility of the classical construction problems
- Corona theorem
- Henselian field
Further reading
- Web https://en.wikipedia.org/wiki/Maximal_ideal, Definition
About this page
| Tag | 09J6KGT65G |
|---|---|
| URI | m2://09J6KGT65G |
| Kind | definition |
| Writing | W1 stated |
| MSC | 13A15 |
| Defines | Maximal ideal |
| Uses | Ideal |
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What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
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