$R\mathrm{/}M$ is a field if and only if $M$ is a maximal ideal

Depends on

• Prime ideals and maximal ideals in a commutative ring
• For a two-sided ideal $I$, the additive cosets form a ring $R/I$ with identity $1+I$
• Every field is a commutative ring with $1 \ne 0$; it is an integral domain, and it is a commutative division ring
• Field
• Ideal criteria and intersections of ideals

Used by

• Every maximal ideal of a commutative ring is prime Corollary
• Over an algebraically closed field, maximal ideals of an affine algebra are kernels of points Corollary
• Rings of integers are Dedekind domains Corollary
• Total ramification at a prime-power cyclotomic level Corollary
• Under an integral extension, a prime is maximal if and only if its contraction is maximal Corollary
• A regular point that is not smooth: a purely inseparable thickening Counterexample
• The equation must define the intended scheme Counterexample
• Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
• A local ring is a nonzero commutative ring with a unique maximal ideal Definition
• Multiplicity of a hypersurface equation at a rational point Definition
• F[x]₍ₓ₎ is the ring of rational functions defined at 0, with maximal ideal generated by x and residue field F Example
• Fitting ideals of a diagonal two-by-two presentation Example
• For every integer n>1, nℤ is a maximal ideal of ℤ if and only if n is prime Example
• S-units of Q Example
• The reals are the quotient of rational Cauchy sequences by the maximal ideal of null sequences Example
• A classical affine algebraic set has a unique finite irredundant decomposition Lemma
• A maximal ideal of an affine algebra has finite residue field over the base field Lemma
• All initial forms define the tangent cone Lemma
• Evaluation at a point has kernel (x₁-a₁,..., xₙ-aₙ) Lemma
• Every central character of a semisimple enveloping algebra arises from a weight Lemma
• Points of an affine algebraic set correspond to maximal ideals of its coordinate ring Lemma
• Tangent vectors at rational points are dual-number points Lemma
• The spectrum of a finite product ring is the disjoint union of the factor spectra Lemma
• A primitive ideal determines a central character Proposition
• A Noetherian ring is Artinian exactly when every prime ideal is maximal Theorem
• Assuming Choice, every field has an algebraic extension containing roots of all nonconstant base polynomials Theorem
• Every commutative Artinian ring is Noetherian Theorem
• Every prime ideal of an Artinian ring is maximal Theorem
• For a nonconstant p in F[x], the ideal (p) is maximal and F[x]/(p) is a field exactly when p is irreducible Theorem
• Ramification is detected by the number-field discriminant Theorem

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Ernst, An Inquiry-Based Approach to Abstract Algebra, Maximal and Prime Ideals (standard reference, not scraped)