The ring spectrum of a commutative ring 
with multiplicative
identity is the set of proper prime
ideals,
 |
(1)
|
The classical example is the spectrum of polynomial rings. For instance,
![Spec(C[x])={<x-a>:a in C} union {<0>},](/images/equations/RingSpectrum/NumberedEquation2.svg) |
(2)
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and
![Spec(C[x,y])={<x-a,y-b>,(a,b) in C^2}
union {<f(x,y)>:f in C[x,y] is irreducible} union {<0>}.](/images/equations/RingSpectrum/NumberedEquation3.svg) |
(3)
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The points are the prime ideals themselves. In classical algebraic geometry,
when 
is the coordinate ring of an affine
variety over an algebraically closed field, a prime ideal defines
an irreducible variety. Note that 
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
 |
(4)
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where 
is a subset of 
. For example,
 |
(5)
|
In this spectrum, every point is a closed point except for 
,
whose topological closure is 
.
