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Elaborating on Katzel's result, Zhai obtained a sharp short interval result for the number of ways of expressing a square-free integer as sums of two squares. In this note we view Zhai's decom- position as one by degree 1 prime ideals in the Gaussian field and extend Zhai's approach to a wider range of quadratic fields etc.  相似文献   
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Imran Anwar  Zahid Raza 《代数通讯》2013,41(11):4698-4704
For a square-free monomial ideal I ? S = k[x 1, x 2,…, x n ], we introduce the notion of quasi-linear quotients. By using the quasi-linear quotients, we give a new algebraic criterion for the shellability of a pure simplicial complex Δ over [n]. Also, we provide a new criterion for the Cohen–Macaulayness of the face ring of a pure simplicial complex Δ. Moreover, we show that the face ring of the spanning simplicial complex (defined in [2 Anwar , I. , Raza , Z. , Kashif , A. Spanning simplicial complexes of uni-cyclic graphs . To appear in Algebra Colloquium . [Google Scholar]]) of an r-cyclic graph is Cohen–Macaulay.  相似文献   
4.
We show that 17.9% of all elliptic curves over Q, ordered by their exponential height, are semistable, and that there is a positive density subset of elliptic curves for which the root numbers are uniformly distributed. Moreover, for any > 1/6 (resp. > 1/12) the set of Frey curves (resp. all elliptic curves) for which the generalized Szpiro Conjecture |(E)| N E 12 is false has density zero. This implies that the ABC Conjecture holds for almost all Frey triples. These results remain true if we use the logarithmic or the Faltings height. The proofs make use of the fibering argument in the square-free sieve of Gouvêa and Mazur. We also obtain conditional as well as unconditional lower bounds for the number of curves with Mordell–Weil rank 0 and 2, respectively.  相似文献   
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Yasser Ibrahim 《代数通讯》2018,46(5):1983-1995
The notion of a U-module was introduced and thoroughly investigated in [11 Ibrahim, Y., Yousif, M. F. (2017). U-modules. Comm. Algebra, doi:https://doi.org/10.1080/00927872.2017.1339064.[Crossref] [Google Scholar]] as a strict and simultaneous generalization of quasi-continuous, square-free and automorphism-invariant modules. In this paper a right R-module M is called a U*-module if every submodule of M is a U-module, and a ring R is called a right U*-ring if RR is a U*-module. We show that M is a U*-module iff whenever A and B are submodules of M with A?B and AB = 0, AB is a semisimple summand of M; equivalently M = XY, where X is semisimple, Y is square-free, and X &; Y are orthogonal. In particular, a ring R is a right U*-ring iff R is a direct product of a square-full semisimple artinian ring and a right square-free ring. Moreover, right U*-rings are shown to be directly-finite, and if the ring is also an exchange ring then it satisfies the substitution property, has stable-range 1, and hence is stably-finite. These results are non-trivial extensions of similar ones on rings all of whose right ideals are either quasi-continuous or auto-invariant.  相似文献   
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A circulant is a Cayley graph of a cyclic group. Arc-transitive circulants of square-free order are classified. It is shown that an arc-transitive circulant of square-free order n is one of the following: the lexicographic product , or the deleted lexicographic , where n = bm and is an arc-transitive circulant, or is a normal circulant, that is, Aut has a normal regular cyclic subgroup.  相似文献   
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Utumi modules     
A right R-module M is called a U-module if, whenever A and B are submodules of M with A?B and AB = 0, there exist two summands K and L of M such that A?essK, B?essL and KL?M. The class of U-modules is a simultaneous and strict generalization of three fundamental classes of modules; namely, the quasi-continuous, the square-free, and the automorphism-invariant modules. In this paper we show that the class of U-modules inherits some of the important features of the aforementioned classes of modules. For example, a U-module M is clean if and only if it has the finite exchange property, if and only if it has the full exchange property. As an immediate consequence, every strongly clean U-module has the substitution property and hence is Dedekind-finite. In particular, the endomorphism ring of a strongly clean U-module has stable range 1.  相似文献   
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Mircea Cimpoeaş 《代数通讯》2018,46(3):1179-1187
Given two finite sequences of positive integers α and β, we associate a square-free monomial ideal Iα,β in a ring of polynomials S, and we recursively compute the algebraic invariants of SIα,β. Also, we give precise formulas in special cases.  相似文献   
9.
A. Dubickas 《Acta Appl Math》2003,78(1-3):99-102
We investigate the asymptotic density of surds, namely, the numbers of the form m 1/d with positive integers m and d such that their height is equal to their metric height among all surds. It is shown that, given an integer d2, this density is equal to 6/2. Moreover, similar results for some fixed m are also obtained.  相似文献   
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