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1.
Let denote the t-class group of an integral domain A. P. Samuel has established that if A is a Krull domain then the mapping , is injective and if A is a regular UFD, then , is bijective. Later, L. Claborn extended this result in case A is a regular Noetherian domain. In the first part of this paper we prove that the mapping ; is an injective homomorphism and in case of an integral domain A such that each υ-invertible υ-ideal of A has υ-finite type, we give an equivalent condition for , to be bijective, thus generalizing the result of Claborn. In the second part of this paper, we define the S-class group of an integral domain A: let S be a (not necessarily saturated) multiplicative subset of an integral domain A. Following [11], a nonzero fractional ideal I of A is S-principal if there exist an and such that . The S-class group of A, S- is the group of fractional t-invertible t-ideals of A under t-multiplication modulo its subgroup of S-principal t-invertible t-ideals of A. We generalize some known results developed for the classic contexts of Krull and PυMD domain and we investigate the case of isomorphism S--. 相似文献
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Mi Hee Park Byung Gyun Kang Phan Thanh Toan 《Journal of Pure and Applied Algebra》2018,222(8):2299-2309
Let be the power series ring over a commutative ring R with identity. For , let denote the content ideal of f, i.e., the ideal of R generated by the coefficients of f. We show that if R is a Prüfer domain and if such that is locally finitely generated (or equivalently locally principal), then a Dedekind–Mertens type formula holds for g, namely for all . More generally for a Prüfer domain R, we prove the content formula for all . As a consequence it is shown that an integral domain R is completely integrally closed if and only if for all nonzero , which is a beautiful result corresponding to the well-known fact that an integral domain R is integrally closed if and only if for all nonzero , where is the polynomial ring over R.For a ring R and , if is not locally finitely generated, then there may be no positive integer k such that for all . Assuming that the locally minimal number of generators of is , Epstein and Shapiro posed a question about the validation of the formula for all . We give a negative answer to this question and show that the finiteness of the locally minimal number of special generators of is in fact a more suitable assumption. More precisely we prove that if the locally minimal number of special generators of is , then for all . As a consequence we show that if is finitely generated (in particular if ), then there exists a nonnegative integer k such that for all . 相似文献
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Let D be a commutative domain with field of fractions K, let A be a torsion-free D-algebra, and let B be the extension of A to a K-algebra. The set of integer-valued polynomials on A is , and the intersection of with is , which is a commutative subring of . The set may or may not be a ring, but it always has the structure of a left -module.A D-algebra A which is free as a D-module and of finite rank is called -decomposable if a D-module basis for A is also an -module basis for ; in other words, if can be generated by and A. A classification of such algebras has been given when D is a Dedekind domain with finite residue rings. In the present article, we modify the definition of -decomposable so that it can be applied to D-algebras that are not necessarily free by defining A to be -decomposable when is isomorphic to . We then provide multiple characterizations of such algebras in the case where D is a discrete valuation ring or a Dedekind domain with finite residue rings. In particular, if D is the ring of integers of a number field K, we show that an -decomposable algebra A must be a maximal D-order in a separable K-algebra B, whose simple components have as center the same finite unramified Galois extension F of K and are unramified at each finite place of F. Finally, when both D and A are rings of integers in number fields, we prove that -decomposable algebras correspond to unramified Galois extensions of K. 相似文献
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Cristhian E. Hidber Miguel A. Xicoténcatl 《Journal of Pure and Applied Algebra》2018,222(6):1478-1488
The purpose of this article is to compute the mod 2 cohomology of , the mapping class group of the Klein bottle with q marked points. We provide a concrete construction of Eilenberg–MacLane spaces and fiber bundles , where denotes the configuration space of unordered q-tuples of distinct points in and is the classifying space of the group . Moreover, we show the mod 2 Serre spectral sequence of the bundle above collapses. 相似文献
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Jan O. Kleppe 《Journal of Pure and Applied Algebra》2018,222(3):610-635
Let be the scheme parameterizing graded quotients of with Hilbert function H (it is a subscheme of the Hilbert scheme of if we restrict to quotients of positive dimension, see definition below). A graded quotient of codimension c is called standard determinantal if the ideal I can be generated by the minors of a homogeneous matrix . Given integers and , we denote by the stratum of determinantal rings where are homogeneous of degrees .In this paper we extend previous results on the dimension and codimension of in to artinian determinantal rings, and we show that is generically smooth along under some assumptions. For zero and one dimensional determinantal schemes we generalize earlier results on these questions. As a consequence we get that the general element of a component W of the Hilbert scheme of is glicci provided W contains a standard determinantal scheme satisfying some conditions. We also show how certain ghost terms disappear under deformation while other ghost terms remain and are present in the minimal resolution of a general element of . 相似文献
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Diogo Diniz Claudemir Fidelis Bezerra Júnior 《Journal of Pure and Applied Algebra》2018,222(6):1388-1404
Let F be an infinite field. The primeness property for central polynomials of was established by A. Regev, i.e., if the product of two polynomials in distinct variables is central then each factor is also central. In this paper we consider the analogous property for and determine, within the elementary gradings with commutative neutral component, the ones that satisfy this property, namely the crossed product gradings. Next we consider , where R admits a regular grading, with a grading such that is a homogeneous subalgebra and provide sufficient conditions – satisfied by with the trivial grading – to prove that has the primeness property if does. We also prove that the algebras satisfy this property for ordinary central polynomials. Hence we conclude that, over a field of characteristic zero, every verbally prime algebra has the primeness property. 相似文献
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Pham Hung Quy 《Journal of Pure and Applied Algebra》2018,222(5):1126-1138
Let be an equidimensional excellent local ring of characteristic . The aim of this paper is to show that does not depend on the choice of parameter ideal provided R is an F-injective local ring that is F-rational on the punctured spectrum. 相似文献
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Dennis I. Merino 《Linear algebra and its applications》2012,436(7):1960-1968
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Gábor Korchmáros Maria Montanucci Pietro Speziali 《Journal of Pure and Applied Algebra》2018,222(7):1810-1826
Let be the algebraic closure of a finite field of odd characteristic p. For a positive integer m prime to p, let be the transcendence degree 1 function field defined by . Let and . The extension is a non-Galois extension. Let K be the Galois closure of F with respect to H. By Stichtenoth [20], K has genus , p-rank (Hasse–Witt invariant) and a -automorphism group of order at least . In this paper we prove that this subgroup is the full -automorphism group of K; more precisely where Δ is an elementary abelian p-group of order and D has an index 2 cyclic subgroup of order . In particular, , and if K is ordinary (i.e. ) then . On the other hand, if G is a solvable subgroup of the -automorphism group of an ordinary, transcendence degree 1 function field L of genus defined over , then ; see [15]. This shows that K hits this bound up to the constant .Since has several subgroups, the fixed subfield of such a subgroup N may happen to have many automorphisms provided that the normalizer of N in is large enough. This possibility is worked out for subgroups of Δ. 相似文献
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Let V be a 6-dimensional vector space over a field , let f be a nondegenerate alternating bilinear form on V and let denote the symplectic group associated with . The group has a natural action on the third exterior power of V and this action defines five families of nonzero trivectors of V. Four of these families are orbits for any choice of the field . The orbits of the fifth family are in one-to-one correspondence with the quadratic extensions of that are contained in a fixed algebraic closure of . In this paper, we divide the orbits corresponding to the separable quadratic extensions into suborbits for the action of on . 相似文献
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Houmem Belkhechine Imed Boudabbous Kaouthar Hzami 《Comptes Rendus Mathematique》2013,351(13-14):501-504
We consider a tournament . For , the subtournament of T induced by X is . An interval of T is a subset X of V such that, for and , if and only if . The trivial intervals of T are ?, and V. A tournament is indecomposable if all its intervals are trivial. For , denotes the unique indecomposable tournament defined on such that is the usual total order. Given an indecomposable tournament T, denotes the set of such that there is satisfying and is isomorphic to . Latka [6] characterized the indecomposable tournaments T such that . The authors [1] proved that if , then . In this note, we characterize the indecomposable tournaments T such that . 相似文献
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A non-exact monotone twist map is a composition of an exact monotone twist map with a generating function H and a vertical translation with . We show in this paper that for each , there exists a critical value depending on H and ω such that for , the non-exact twist map has an invariant Denjoy minimal set with irrational rotation number ω lying on a Lipschitz graph, or Birkhoff -periodic orbits for rational . Like the Aubry–Mather theory, we also construct heteroclinic orbits connecting Birkhoff periodic orbits, and show that quasi-periodic orbits in these Denjoy minimal sets can be approximated by periodic orbits. In particular, we demonstrate that at the critical value , the Denjoy minimal set is not uniformly hyperbolic and can be approximated by smooth curves. 相似文献
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One of the most frequently referenced monographs on power series rings, “Power Series over Commutative Rings” by James W. Brewer, states in Theorem 21 that if M is a non-SFT maximal ideal of a commutative ring R with identity, then there exists an infinite ascending chain of prime ideals in the power series ring , such that for each n. Moreover, the height of is infinite. In this paper, we show that the above theorem is false by presenting two counter examples. The first counter example shows that the height of can be zero (and hence there is no chain of prime ideals in satisfying for each n). In this example, the ring R is one-dimensional. In the second counter example, we prove that even if the height of is uncountably infinite, there may be no infinite chain of prime ideals in satisfying for each n. 相似文献
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