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**1 - 2**of**2**### Do Noetherian Modules Have Noetherian Basis Functions?

"... Abstract. In Bishop-style constructive algebra it is known that if a module over a commutative ring has a Noetherian basis function, then it is Noetherian. Using countable choice we prove the reverse implication for countable and strongly discrete modules. The Hilbert basis theorem for this specific ..."

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Abstract. In Bishop-style constructive algebra it is known that if a module over a commutative ring has a Noetherian basis function, then it is Noetherian. Using countable choice we prove the reverse implication for countable and strongly discrete modules. The Hilbert basis theorem for this specific class of Noetherian modules, and polynomials in a single variable, follows with Tennenbaum’s celebrated version for modules with a Noetherian basis function. In particular, the usual hypothesis that the modules under consideration are coherent need not be made. We further identify situations in which countable choice is dispensable. 1

### Spectral Schemes as Ringed Lattices

, 2009

"... “ What would have happened if topologies without points had been discovered before topologies with points, or if Grothendieck had known the theory of dis-tributive lattices? ” Gian-Carlo Rota, Indiscrete Thoughts. Birkhäuser (1997), p. 220 We give a point-free definition of a Grothendieck scheme wh ..."

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“ What would have happened if topologies without points had been discovered before topologies with points, or if Grothendieck had known the theory of dis-tributive lattices? ” Gian-Carlo Rota, Indiscrete Thoughts. Birkhäuser (1997), p. 220 We give a point-free definition of a Grothendieck scheme whose underlying topo-logical space is spectral. Affine schemes aside, the prime examples are the projective nonsingular curve. With the appropriate notion of a morphism between spectral schemes, elementary proofs of the universal properties become possible.