Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
Journal Prestige (SJR): 0.339 Number of Followers: 1 Partially Free Journal ISSN (Print) 01384821  ISSN (Online) 21910383 Published by SpringerVerlag [2467 journals] 
 Channel linear Weingarten surfaces in space forms

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Abstract: Abstract Channel linear Weingarten surfaces in space forms are investigated in a Lie sphere geometric setting, which allows for a uniform treatment of different ambient geometries. We show that any channel linear Weingarten surface in a space form is isothermic and, in particular, a surface of revolution in its ambient space form. We obtain explicit parametrisations for channel surfaces of constant Gauss curvature in space forms, and thereby for a large class of linear Weingarten surfaces up to parallel transformation.
PubDate: 20230123

 On purelymaximal ideals and semiNoetherian power series rings

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Abstract: Abstract Tarizadeh and Aghajani conjectured that each purelyprime ideal is purelymaximal (Tarizadeh and Aghajani in Commun Algebra 49(2):824–835, 2021, Conjecture 5.8). We study purelyprime and purelymaximal ideals in rings of the form \(A+XS\) (where S is either B[X] or B[[X]]), subrings of A[[X]] of the form \(A[X]+I[[X]]\) and \(A+I[[X]]\) (where A is a subring of a commutative unitary ring B and I an ideal of A) and Nagata’s idealization ring. As application, we give necessary and sufficient conditions on each of the aforementioned ring to be semiNoetherian. We deduce that the power series ring A[[X]] is semiNoetherian if and only if the ring A is semiNoetherian. We deduce that Tarizadeh and Aghajani’s conjecture holds in each of the aforementioned ring if and only if it holds in the ring A.
PubDate: 20230123

 Groebner fans and embedded resolutions of ideals on toric varieties

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Abstract: Abstract We consider the notions of Groebner fan and Newton nondegeneracy for an ideal on a toric variety, extending the two existing notions for ideals on affine spaces. We prove, without assumptions on the characteristic of the base fields, that the “Groebner fan” of such an ideal is actually a polyhedral fan and that a subvariety defined by a Newton nondegenerate ideal on a toric variety \(X_\sigma \) admits a toric embedded resolution of singularities \(Z\longrightarrow X_\sigma .\)
PubDate: 20230118

 Permutation identities and fractal structure of rings

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Abstract: Abstract We introduce the notion of a product fractal ideal of a ring using permutations of finite sets and multiplication operation in the ring. This notion generalizes the concept of an ideal of a ring. We obtain the corresponding quotient structure that partitions the ring under certain conditions. We prove fractal isomorphism theorems and illustrate the fractal structure involved with examples. These fractal isomorphism theorems extend the classical isomorphism theorems in rings, providing a broader viewpoint.
PubDate: 20230109

 Nearly Kaehler manifolds admitting a closed conformal vector field

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Abstract: Abstract We study a nearly Kaehler manifold M admitting a closed conformal vector field V, and obtain three results under the following assumptions (i) V is almost analytic, (ii) M has real dimension \(>6\) , is complete and strictly nearly Kaehler, and (iii) M is complete strictly nearly Kaehler of global constant type.
PubDate: 20230109

 On the source algebra equivalence class of blocks with cyclic defect
groups, I
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Abstract: Abstract We investigate the source algebra class of a pblock with cyclic defect groups of the group algebra of a finite group. By the work of Linckelmann this class is parametrized by the Brauer tree of the block together with a sign function on its vertices and an endopermutation module of a defect group. We prove that this endopermutation module can be read off from the character table of the group. We also prove that this module is trivial for all cyclic pblocks of quasisimple groups with a simple quotient which is a sporadic group, an alternating group, a group of Lie type in defining characteristic, or a group of Lie type in crosscharacteristic for which the prime p is large enough in a certain sense.
PubDate: 20230104

 Infinitesimal Torelli for weighted complete intersections and certain Fano
threefolds
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Abstract: Abstract We generalize the classical approach of describing the infinitesimal Torelli map in terms of multiplication in a Jacobi ring to the case of quasismooth complete intersections in weighted projective space. As an application, we prove that the infinitesimal Torelli theorem does not hold for hyperelliptic Fano threefolds of Picard rank 1, index 1, degree 4, and study the action of the automorphism group on cohomology. The results of this paper are used to prove LangVojta’s conjecture for the moduli of such Fano threefolds in a followup paper.
PubDate: 20230102

 Helical surfaces with a constant ratio of principal curvatures

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Abstract: Abstract We determine all helical surfaces in threedimensional Euclidean space which possess a constant ratio \(a:=\kappa _1/\kappa _2\) of principal curvatures (CRPC surfaces), thus providing the first explicit CRPC surfaces beyond the known rotational ones. Our approach is based on the involution of conjugate surface tangents and on well chosen generating profiles such that the characterizing differential equation is sufficiently simple to be solved explicitly. We analyze the resulting surfaces, their behavior at singularities that occur for \(a>0\) , and provide an overview of the possible shapes.
PubDate: 20221224

 Bounded birationality and isomorphism problems are computable

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Abstract: Abstract Let X, Y be two irreducible subvarieties of the projective space \({\mathbb {P}}^n\) , and \(d\ge 1\) an integer number. The main result of this paper is an algorithm to construct explicitly, in terms of d and the ideals defining X and Y, a quasiaffine algebraic variety parametrising the set of all birational maps f from X onto Y which can be extended to a selfrational map of \({\mathbb {P}}^n\) of algebraic degree \(\le d\) . We also prove similar results for the case f is a dominant rational map, regular morphism, isomorphism or regular embedding. Similar results are valid for varieties over an arbitrary algebraically closed field, and also for maps on nonprojective varieties.
PubDate: 20221221

 Quadrilateral reptiles

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Abstract: Abstract A polygon P is called a reptile, if it can be decomposed into \(k\ge 2\) nonoverlapping and congruent polygons similar to P. We prove that if a cyclic quadrilateral is a reptile, then it is a trapezoid. Comparing with results of Betke and Osburg we find that every convex reptile is a triangle or a trapezoid.
PubDate: 20221219

 A norm functor for quadratic algebras

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Abstract: Abstract Given commutative, unital rings \(\mathcal {A}\) and \(\mathcal {B}\) with a ring homomorphism \(\mathcal {A}\rightarrow \mathcal {B}\) making \(\mathcal {B}\) free of finite rank as an \(\mathcal {A}\) module, we can ask for a “trace” or “norm” homomorphism taking algebraic data over \(\mathcal {B}\) to algebraic data over \(\mathcal {A}\) . In this paper we we construct a norm functor for the data of a quadratic algebra: given a locallyfree rank2 \(\mathcal {B}\) algebra \(\mathcal {D}\) , we produce a locallyfree rank2 \(\mathcal {A}\) algebra \(\textrm{Nm}_{\mathcal {B}/\mathcal {A}}(\mathcal {D})\) in a way that is compatible with other norm functors and which extends a known construction for étale quadratic algebras. We also conjecture a relationship between discriminant algebras and this new norm functor.
PubDate: 20221202

 Absolute Ideals of Murley Groups

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Abstract: Abstract For an Abelian group G, a subgroup A of G is called an absolute ideal of G if A is an ideal of any ring on G. If R is a ring and any ideal of R is an absolute ideal of the additive group of R, then R is called an AI ring. If G is an Abelian group and there exists an AIring on G, then G is called an RAIgroup. For RAIgroups, the description problem is formulated by L. Fuchs. Obviously, every full invariant subgroup of an Abelian group G is an absolute ideal of G. E. Fried formulated the problem of studying Abelian groups for which the converse is true; i.e., every absolute ideal is a fully invariant subgroup. Such groups are called afi groups. In this work, we describe absolute ideals of Murley groups. This allows us to describe RAIgroups, afigroups, and Egroups in the class of Murley groups.
PubDate: 20221201
DOI: 10.1007/s13366021006040

 The geometric reason for the nonexistence of a MOL(6)

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Abstract: Abstract The problem of Euler concerning the 36 officers, (Euler, in Leonardi Euleri Opera Ser I 7:291–392, 1782), was first solved by Tarry (Comptes rendus Ass Franc Sci Nat 1 (1900) 2:170–203, 1901). Short proofs for the nonexistence were given in Betten (Unterricht 36:449–453, 1983), Beth et al. (Design Theory, Bibl. Inst. Mannheim, Wien, Zürich, 1985), Stinson (J Comb Theory A 36:373–376, 1984). This problem is equivalent to the existence of a MOL(6), i. e., a pair of mutually orthogonal latin squares of order 6. Therefore in Betten (Mitt Math Ges Hamburg 39:59–76, 2019; Res Math 76:9, 2021; Algebra Geom 62:815–821, 2021) the structure of a (hypothetical) MOL(6) was studied. Now we combine the old proofs and the new studies and filter out a simple way for the proof of nonexistence. The aim is not only to give still other short proofs, but to analyse the problem and reveal the geometric reason for the nonexistence of a MOL(6) and the nonsolvability of Euler’s problem.
PubDate: 20221201
DOI: 10.1007/s13366021006188

 On modules satisfying Sdccr condition

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Abstract: Abstract In this paper, we introduce a new class of modules satisfying Sdccr (Sdccr \(^{\star })\) condition which is a generalization of Sartinian modules. Let \(A\ \) be a commutative ring with \(0\ne 1\ \) and \(X\ \) a unital Amodule. Suppose that \(S\subseteq A\ \) is a multiplicatively closed subset. \(X\ \) is said to satisfy Sdccr (Sdccr \(^{\star })\) condition if for each finitely generated (principal) ideal \(I\ \) of \(A\ \) and a submodule \(Y\ \) of \(X,\ \) the descending chain \(\{I^{i}Y\}_{i\in {\mathbb {N}}}\) is Sstationary. Many examples and properties of modules satisfying Sdccr (Sdccr \(^{\star })\ \) condition are given. Furthermore, we characterize modules satisfying dccr (dccr \(^{\star })\) condition in terms of some known class of rings and modules. Also, we give Nakayama’s Lemma for modules satisfying Sdccr condition.
PubDate: 20221201
DOI: 10.1007/s13366021006099

 On the existence of four or more curved foldings with common creases and
crease patterns
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Abstract: Abstract Consider an oriented curve \(\Gamma \) in a domain D in the plane \({\varvec{R}}^2\) . Thinking of D as a piece of paper, one can make a curved folding in the Euclidean space \({\varvec{R}}^3\) . This can be expressed as the image of an “origami map” \(\Phi :D\rightarrow {\varvec{R}}^3\) such that \(\Gamma \) is the singular set of \(\Phi \) , the word “origami” coming from the Japanese term for paper folding. We call the singular set image \(C:=\Phi (\Gamma )\) the crease of \(\Phi \) and the singular set \(\Gamma \) the crease pattern of \(\Phi \) . We are interested in the number of origami maps whose creases and crease patterns are C and \(\Gamma \) , respectively. Two such possibilities have been known. In the authors’ previous work, two other new possibilities and an explicit example with four such noncongruent distinct curved foldings were established. In this paper, we determine the possible values for the number N of congruence classes of curved foldings with the same crease and crease pattern. As a consequence, if C is a nonclosed simple arc, then \(N=4\) if and only if both \(\Gamma \) and C do not admit any symmetries. On the other hand, when C is a closed curve, there are infinitely many distinct possibilities for curved foldings with the same crease and crease pattern, in general.
PubDate: 20221201
DOI: 10.1007/s13366021006022

 On Auslander–Reiten components of string complexes for a certain class
of symmetric special biserial algebras
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Abstract: Abstract Let \(\mathbb {k}\) be an algebraically closed field. In this article, inspired by the description of indecomposable objects in the derived category of a gentle algebra obtained by Bekkert and Merklen, we define string complexes for a certain class \({\mathscr {C}}\) of symmetric special biserial algebras, which are indecomposable perfect complexes in the corresponding derived category. We also prove that if \(\Lambda \) is a \(\mathbb {k}\) algebra in the class \({\mathscr {C}}\) and \(P^\bullet \) is a string complex over \(\Lambda \) , then \(P^\bullet \) lies in the rim of its Auslander–Reiten component.
PubDate: 20221201
DOI: 10.1007/s1336602100607x

 Constructing cubic curves with involutions

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Abstract: Abstract In 1888, Heinrich Schroeter provided a ruler construction for points on cubic curves based on line involutions. Using Chasles’ Theorem and the terminology of elliptic curves, we give a simple proof of Schroeter’s construction. In addition, we show how to construct tangents and additional points on the curve using another ruler construction which is also based on line involutions. As an application of Schroeter’s construction we provide a new parametrisation of elliptic curves with torsion group \(\mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/8\mathbb {Z}\) and give some configurations with all their points on a cubic curve.
PubDate: 20221201
DOI: 10.1007/s13366021005930

 Minimal crystallizations of 3manifolds with boundary

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Abstract: Abstract Let \((\Gamma ,\gamma )\) be a crystallization of connected compact 3manifold M with h boundary components. Let \(\mathcal {G}(M)\) and \( k (M)\) be the regular genus and gemcomplexity of M respectively, and let \(\mathcal {G}(\partial M)\) be the regular genus of \(\partial M\) . We prove that $$\begin{aligned} k (M)\ge 3 (\mathcal {G}(M)+h1) \ge 3 (\mathcal {G} (\partial M)+h1). \end{aligned}$$ These bounds for gemcomplexity of M are sharp for several 3manifolds with boundary. Further, we show that if \(\partial M\) is connected and \( k (M)< 3 (\mathcal {G} (\partial M)+1)\) then M is a handlebody. In particular, we prove that \( k (M) =3 \mathcal {G} (\partial M)\) if M is a handlebody and \( k (M) \ge 3 (\mathcal {G} (\partial M)+1)\) if M is not a handlebody. Further, we obtain several combinatorial properties for a crystallization of 3manifolds with boundary.
PubDate: 20221201
DOI: 10.1007/s13366021005989

 A note on modular Terwilliger algebras of association schemes

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Abstract: Abstract Let p denote a prime number. In this note, we focus on the modular Terwilliger algebras of association schemes defined in Hanaki (Graphs Combin, 2021, https://doi.org/10.1007/s00373021023630). We define the primary module of a modular Terwilliger algebra of an association scheme and determine all its composition factors up to isomorphism. We then characterize the \(p'\) valenced association schemes in terms of numerous properties of their modular Terwilliger algebras. We conclude our investigation with a few corollaries and questions on the modular Terwilliger algebras of association schemes.
PubDate: 20221201
DOI: 10.1007/s1336602100605z

 Nonnil–Laskerian rings

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Abstract: Abstract Let R be a commutative ring with unity. In this paper we introduce the concept of Nonnil–Laskerian ring that is related to the class of Laskerian rings. A ring R is said to be Nonnil–Laskerian if every nonnil ideal I of R is decomposable. We show that Nonnil–Laskerian rings enjoy analogs of many properties of Laskerian ring. We give an example of Nonnil–Laskerian ring, wich is not Laskerian. We study the Nonnil–Laskerian property over the polynomial and formel power series rings. In particular, we show that we have not an equivalence between Nonnil–Laskerian and Nonnil–Noetherian concepts in R[[X]] and R[X], contrary to the Laskerian and Noetherian concepts.
PubDate: 20221201
DOI: 10.1007/s13366021006031
