Space of Valuations
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چکیده
We present a constructive interpretation of some results in [6]. To any integral domain R we associate a distributive lattice V (R) which is a point-free presentation of the space of valuations over the ring R. Our definition, by generators and relations, is similar to Joyal’s definition of the Zariski lattice of a ring R, which is a point-free presentation of the Zariski spectrum of R. The space represented by V (R) is also called the abstract Riemann surface associated to R [8]. There is a natural map φ : Z(R)→ V (R) from the Zariski lattice of R to the lattice V (R), and correspond to the usual map V 7−→ R ∩ mV which associates to any valuation ring V its center on R. This map is always surjective, which in term of lattices means that φ satisfies that φ(u) ≤ φ(v) is equivalent to u ≤ v. If R is a Prüfer ring, then φ is bijective. In [2, 3] we have developped a constructive theory of the Krull dimension of rings and distributive lattices. We can then define then the valuative dimension Vdim R of R to be the Krull dimension of V (R). We show constructively that the map φ has the going-up property and one deduces from this that one has always Kdim R ≤ Vdim R. We give also then a point-free proof that Vdim R[X] ≤ 1+Vdim R. An application of these results is to provide a constructive proof of Kdim R[X1, . . . , Xn] = n+ Kdim R is R is a Prüfer ring.
منابع مشابه
Quotient BCI-algebras induced by pseudo-valuations
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