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41.
François Couchot 《代数通讯》2013,41(10):3418-3423
It is proved that localizations of injective R-modules of finite Goldie dimension are injective if R is an arithmetical ring satisfying the following condition: for every maximal ideal P, R P is either coherent or not semicoherent. If, in addition, each finitely generated R-module has finite Goldie dimension, then localizations of finitely injective R-modules are finitely injective too. Moreover, if R is a Prüfer domain of finite character, localizations of injective R-modules are injective. 相似文献
42.
Yang Zhang 《代数通讯》2013,41(1):373-381
Some necessary and sufficient conditions are given for rings of the Morita contexts to be semihereditary maximal V-orders, Prüfer rings and Dubrovin valuation rings. 相似文献
43.
M. Chacron 《代数通讯》2013,41(11):4613-4631
Let D be a division ring with centre Z and with involution (*). Let V be a valuation of D with value group Γ, a linearly ordered additive group (non necessarily commutative) together with a symbol ∞ (positive infinity). We assume that for each nonzero symmetric element s = s* of D, which is algebraic over Z, we have for all nonzero elements x of D, V(xa ? ax) > V(ax). We define the residue characteristic exponent p of V to be the characteristic χ of the associated residue division ring written as D V , if χ ≠ 0, and p = 1, if χ = 0. We show here that if F is a finite dimensional commutative subalgebra of D over Z, which is *-closed (i.e., F* = F), and if (*) is of the first kind (i.e., each central element of D must be symmetric), then [F: Z] = 2 r p m where m is a nonnegative integer and r = 0 or 1 according as the restricted involution in F is trivial or not. The case of an involution (*) of the second kind (i.e., some central element of D is not symmetric) requires (for this author) a stronger type of valuation, namely, V is a *-valuation, that is to say, for all elements x of D, we have V(x*) = V(x), a condition which readily implies Γ must be Abelian. Here, we can show that for F as in the preceding, [F: Z] = p m , where m is again a nonnegative integer. The preceding results generalize a theorem of Gräter and improve in parts recent theorems of this author in [2]. In the special case p = 2 the results provide a modicum of answers to the questions opened informally in [2] (see concluding paragraph in [2] or here Question 3.2.1). More is to be said in the third and final section of this work. 相似文献
44.
Steven Dale Cutkosky 《代数通讯》2013,41(7):2828-2866
In characteristic zero, local monomialization is true along any valuation. However, we have recently shown that local monomialization is not always true in positive characteristic, even in two dimensional algebraic function fields. In this paper we show that local monomialization is true for defectless extensions of two dimensional excellent local rings, extending an earlier result of Piltant and the author for two dimensional algebraic function fields over an algebraically closed field. We also give theorems showing that in many cases there are good stable forms of the extension of associated graded rings in a finite separable field extension. 相似文献
45.
John S. Kauta 《代数通讯》2013,41(11):3566-3589
A nonassociative quaternion algebra over a field F is a 4-dimensional F-algebra A whose nucleus is a separable quadratic extension field of F. We define the notion of a valuation ring for A, and we also define a value function on A with values from a totally ordered group. We determine the structure of the set on which the function assumes non-negative values and we prove that, given a valuation ring of A, there is a value function associated to it if and only if the valuation ring is integral and invariant under proper F-automorphisms of A. 相似文献
46.
47.
In this article we study rank one discrete valuations of the field k((X 1,…, X n )) whose center in k[[X 1,…, X n ]] is the maximal ideal. In Sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote Section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in Section 5. The constructions given in these sections are not effective in the general case, because we need either to use Zorn's lemma or to know explicitly a section σ of the natural homomorphism R v → Δ v between the ring and the residue field of the valuation v. However, as a consequence of this construction, in Section 7, we prove that k((X 1,…, X n )) can be embedded into a field L((Y 1,…, Y n )), where L is an algebraic extension of k and the “extended valuation” is as close as possible to the usual order function. 相似文献
48.
《Journal of Pure and Applied Algebra》2022,226(7):106898
The almost purity theorem is central to the geometry of perfectoid spaces and has numerous applications in algebra and geometry. This result is known to have several different proofs in the case that the base ring is a perfectoid valuation ring. We give a new proof by exploiting the behavior of Faltings' normalized length under the Frobenius map. 相似文献
49.
《Journal of Pure and Applied Algebra》2022,226(10):107088
We describe the structure of projective covers of modules over a local ring, when such covers exist, and modules with minimal sets of generators. The case of modules over valuation rings is studied in more detail. 相似文献
50.