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2008-05-12T16:34:55Z

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Vipul: New page: {{spectrum topology fact}} ==Statement== The spectrum of an integral domain is an irreducible space: it cannot be expressed as a union of two proper closed subsets. ==Conver...

2008-03-15T21:30:03Z

New page: {{spectrum topology fact}} ==Statement== The spectrum of an integral domain is an irreducible space: it cannot be expressed as a union of two proper closed subsets. ==Conver...

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{{spectrum topology fact}}

==Statement==

The [[spectrum]] of an [[integral domain]] is an [[irreducible space]]: it cannot be expressed as a union of two proper closed subsets.

==Converse==

The converse is not in general true. For instance, for a [[local Artinian ring]], the spectrum is just a single point, which ''is'' irreducible, but the ring is not in general an integral domain.

More generally, the spectrum of a ring is irreducible if and only if there is a unique minimal prime ideal, or equivalently, if the [[nilradical]] is a [[prime ideal]].
==Proof==

In fact, the spectrum is the closure of a ''single'' point: the prime ideal <math>0</math>, and hence, it ''must'' be irreducible.

Spectrum of integral domain is irreducible - Revision history

Vipul: 1 revision

Vipul: New page: {{spectrum topology fact}} ==Statement== The spectrum of an integral domain is an irreducible space: it cannot be expressed as a union of two proper closed subsets. ==Conver...