Prime ideal structure in commutative rings

  title={Prime ideal structure in commutative rings},
  author={Melvin Hochster},
  journal={Transactions of the American Mathematical Society},
  • M. Hochster
  • Published 1 August 1969
  • Mathematics
  • Transactions of the American Mathematical Society
0. Introduction. Let ' be the category of commutative rings with unit, and regard Spec (as in [1]) as a contravariant functor from ' to g$7 the category of topological spaces and continuous maps. The spaces of the form Spec A, A a ring (ring always means object of '), are well known to have many special properties. It is worthwhile to discover whether these well-known properties of the spaces characterize them, since we then will know the limitations of the topological approach. As a by-product… 


0. Introduction. Let <€ be the category of commutative rings with unit, and regard Spec (as in [1]) as a contra variant functor from # to 3~, the category of topological spaces and continuous maps.

The Minimal Prime Spectrum of a Commutative Ring

  • M. Hochster
  • Mathematics
    Canadian Journal of Mathematics
  • 1971
We call a topological space X minspectral if it is homeomorphic to the space of minimal prime ideals of a commutative ring A in the usual (hull-kernel or Zariski) topology (see [2, p. 111]). Note

Separation axioms and the prime spectrum of commutative semirings

In this work by a semiring we mean a commutative semiring with nonzero identity and we study the prime spectrum of a semiring with its Zariski topology. We prove that this space is spectral and we

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E. Enochs has defined a ring A (all rings are commutative, with 1) to be totally integrally closed (which we abbreviate TIC) if for every integral extension h: B—>C the induced map /ι*: Horn (C, A)

Zariski-Like Topology on the Classical Prime Spectrum of a Modules

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Let R be a ring, M be a left R-module and Spec(RM) be the collection of all prime submodules of M. In this paper and its sequel, we in- troduce and study a generalization of the Zariski topology of

Algebraic and topological aspects of quasi-prime ideals

In this paper, we define the new notion of quasi-prime ideal which generalizes at once both prime ideal and primary ideal notions. Then a natural topology on the set of quasi-prime ideals of a ring


Rings of continuous functions

Contents: Functions of a Topological Space.- Ideals and Z-Filters.- Completely Regular Spaces.- Fixed Ideals. Compact Spaces.- Ordered Residue Class Rings.- The Stone-Czech Compactification.-