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Spectrum of a ring

The set of all prime ideals of a commutative ring typeset structure with identity is called prime spectrum of typeset structure, and is denoted by typeset structure,

Let typeset structure be commutative ring with identity. Let typeset structure be a subset of typeset structure. Let typeset structure be the set of all prime ideals typeset structure of typeset structure such that typeset structure.

We have:

  • typeset structure, typeset structure
  • if typeset structure are two ideals of typeset structure, then typeset structure
  • if typeset structure is a system of subsets of typeset structure, then typeset structure,
  • if the ideal typeset structure is generated by a set typeset structure, then typeset structure, where typeset structure is the radical of typeset structure.

    • These properties show that the elements of typeset structure satisfy defining axioms of closed sets of a topological space. The corresponding topology on typeset structure is called Zariski topology.1

    Notes

    1 Oscar Zariski (1899-1986) was one of the most influential algebraic geometers of the 20th century ,

    Cite this web-page as:

    Štefan Porubský: Spectrum.

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