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Ring Spectrum


The spectrum of a ring is the set of proper prime ideals,

 Spec(R)={p:p is a prime ideal in R}.
(1)

The classical example is the spectrum of polynomial rings. For instance,

 Spec(C[x])={<x-a>:a in C} union {<0>},
(2)

and

 Spec(C[x,y])={<x-a,y-b>,(a,b) in C^2} 
  union {<f(x,y)>:f is irreducible} union {<0>}.
(3)

The points are, in classical algebraic geometry, algebraic varieties. Note that <x-a,y-b> are maximal ideals, hence also prime.

The spectrum of a ring has a topology called the Zariski topology. The closed sets are of the form

 V(S)={<p>:S subset <p>}.
(4)

For example,

 Spec(Z)={<p>:p is prime} union {<0>}.
(5)

Every prime ideal is closed except for <0>, whose closure is V(0)=Spec(Z).


See also

Affine Scheme, Category Theory, Commutative Algebra, Conic Section, Ideal, Prime Ideal, Projective Algebraic Variety, Scheme, Variety, Zariski Topology

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Ring Spectrum." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/RingSpectrum.html

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