Subtractive Bases of Zariski Spaces

• Autores: R. L. McCasland, Matthew E. Moore, Patrick F. Smith
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 32, Nº 4, 2006, págs. 971-983
• Idioma: inglés
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• Resumen

  • The notion of a subtractive basis for a Zariski Space is defined and examined. Such bases provide a means of generating Zariski Spaces, which exploits both the algebraic and topological-type properties of these spaces. In particular, it is shown that for every finitely-generated module M over a commutative ring with identity, then the Zariski Space of M has a (finite) subtractive basis. Moreover, it is shown that, provided M has the additional property of being a radical module (i.e., rad 0 = 0), then every subtractive basis of the Zariski Space of M corresponds to a direct sum decomposition of M. For such a module M, it then follows that every direct summand A of M must have the following property: if B is a submodule of M and rad B = A, then B = A.