scholarly journals Generating Adjoint Groups

2019 ◽  
Vol 62 (3) ◽  
pp. 733-738 ◽  
Author(s):  
Be'eri Greenfeld
Keyword(s):  
Open Problem ◽  
Jacobson Radical ◽  
The Other ◽  
Adjoint Group ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Other Hand ◽  
Nil Ring

AbstractWe prove two approximations of the open problem of whether the adjoint group of a non-nilpotent nil ring can be finitely generated. We show that the adjoint group of a non-nilpotent Jacobson radical cannot be boundedly generated and, on the other hand, construct a finitely generated, infinite-dimensional nil algebra whose adjoint group is generated by elements of bounded torsion.

scholarly journals On Primitive Ideals in Graded Rings

2008 ◽  
Vol 51 (3) ◽  
pp. 460-466 ◽  
Author(s):  
Agata Smoktunowicz
Keyword(s):  
Prime Ideal ◽  
Jacobson Radical ◽  
Graded Rings ◽  
Dimensional Algebra ◽  
Infinite Dimensional ◽  
Primitive Ideals ◽  
Nil Ring

AbstractLet R = be a graded nil ring. It is shown that primitive ideals in R are homogeneous. Let A = be a graded non-PI just-infinite dimensional algebra and let I be a prime ideal in A. It is shown that either I = ﹛0﹜ or I = A. Moreover, A is either primitive or Jacobson radical.

scholarly journals Zero-sum subsets of decomposable sets in Abelian groups

2020 ◽  
Vol 30 (1) ◽  
pp. 15-25
Author(s):  
T. Banakh ◽  
◽  
A. Ravsky ◽  
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Open Problem ◽  
Abelian Groups ◽  
The Other ◽  
Other Hand ◽  
Decomposable Sets ◽  
Empty Subset ◽  
Zero Sum

A subset D of an abelian group is decomposable if ∅≠D⊂D+D. In the paper we give partial answers to an open problem asking whether every finite decomposable subset D of an abelian group contains a non-empty subset Z⊂D with ∑Z=0. For every n∈N we present a decomposable subset D of cardinality |D|=n in the cyclic group of order 2n−1 such that ∑D=0, but ∑T≠0 for any proper non-empty subset T⊂D. On the other hand, we prove that every decomposable subset D⊂R of cardinality |D|≤7 contains a non-empty subset T⊂D of cardinality |Z|≤12|D| with ∑Z=0. For every n∈N we present a subset D⊂Z of cardinality |D|=2n such that ∑Z=0 for some subset Z⊂D of cardinality |Z|=n and ∑T≠0 for any non-empty subset T⊂D of cardinality |T|<n=12|D|. Also we prove that every finite decomposable subset D of an Abelian group contains two non-empty subsets A,B such that ∑A+∑B=0.

scholarly journals The Complexity of Classification Problems for Models of Arithmetic

2010 ◽  
Vol 16 (3) ◽  
pp. 345-358 ◽  
Author(s):  
Samuel Coskey ◽  
Roman Kossak
Keyword(s):  
Classification Problem ◽  
The Other ◽  
Saturated Model ◽  
Classification Problems ◽  
Finitely Generated ◽  
Saturated Models ◽  
Other Hand ◽  
Recursively Saturated ◽  
Recursively Saturated Model ◽  
Models Of Arithmetic

AbstractWe observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.

scholarly journals Maximal perpendicularity in certain Abelian groups

2017 ◽  
Vol 9 (1) ◽  
pp. 235-247
Author(s):  
Mika Mattila ◽  
Jorma K. Merikoski ◽  
Pentti Haukkanen ◽  
Timo Tossavainen
Keyword(s):  
Abelian Group ◽  
Vector Space ◽  
Binary Relation ◽  
Abelian Groups ◽  
The Other ◽  
Finitely Generated ◽  
Other Hand

AbstractWe define perpendicularity in an Abelian group G as a binary relation satisfying certain five axioms. Such a relation is maximal if it is not a subrelation of any other perpendicularity in G. A motivation for the study is that the poset (𝒫, ⊆) of all perpendicularities in G is a lattice if G has a unique maximal perpendicularity, and only a meet-semilattice if not. We study the cardinality of the set of maximal perpendicularities and, on the other hand, conditions on the existence of a unique maximal perpendicularity in the following cases: G ≅ ℤn, G is finite, G is finitely generated, and G = ℤ ⊕ ℤ ⊕ ⋯. A few such conditions are found and a few conjectured. In studying ℝn, we encounter perpendicularity in a vector space.

scholarly journals RADIKAL SUPERNILPOTENT BERTINGKAT

2019 ◽  
Vol 2 (1) ◽  
pp. 1
Author(s):  
Puguh Wahyu Prasetyo
Keyword(s):  
Jacobson Radical ◽  
Ring Theory ◽  
The Other ◽  
Graded Rings ◽  
Radical Theory ◽  
Other Hand

The development of Ring Theory motivates the existence of the development of the Radical Theory of Rings. This condition is motivated since there are rings which have properties other than those owned by the set ring of all integers. These rings are collected so that they fulfill certain properties and they are called radical classes of rings. As the development of science about how to separate the properties of radical classes of rings motivates the existence of supernilpotent radical classes. On the other hand, there exists the concept of graded rings. This concept can be generalized into the Radical Theory of Rings. Thus, the properties of the graded supernilpotent radical classes are very interesting to investigate. In this paper, some graded supernilpotent radical of rings are given and their construction will be described. It follows from this construction that the graded Jacobson radical is a graded supernilpotent radical.

THE NUMBER OF PROFINITE GROUPS WITH A SPECIFIED SYLOW SUBGROUP

2015 ◽  
Vol 99 (1) ◽  
pp. 108-127
Author(s):  
COLIN D. REID
Keyword(s):  
Normal Subgroup ◽  
Sylow Subgroup ◽  
The Other ◽  
Profinite Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Other Hand ◽  
Formula Group ◽  
Nontrivial Normal Subgroup

Let $S$ be a finitely generated pro-$p$ group. Let ${\mathcal{E}}_{p^{\prime }}(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects nontrivially with every nontrivial normal subgroup of $G$. In this paper, we investigate whether or not there is a bound on $|G:S|$ for $G\in {\mathcal{E}}_{p^{\prime }}(S)$. For instance, we give an example where ${\mathcal{E}}_{p^{\prime }}(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $|G:S|$ is bounded in the case where $S$ is just infinite.

scholarly journals On the Stanley Depth of Powers of Monomial Ideals

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 607 ◽  
Author(s):  
S. A. Seyed Fakhari
Keyword(s):  
Polynomial Ring ◽  
Upper Bound ◽  
Integral Closure ◽  
Monomial Ideals ◽  
The Other ◽  
Finitely Generated ◽  
Stanley Depth ◽  
Survey Article ◽  
Symbolic Powers ◽  
Other Hand

In 1982, Stanley predicted a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The predicted invariant is now called the Stanley depth. Duval et al. found a counterexample for Stanley’s conjecture, and their counterexample is a quotient of squarefree monomial ideals. On the other hand, there is evidence showing that Stanley’s inequality can be true for high powers of monomial ideals. In this survey article, we collect the recent results in this direction. More precisely, we investigate the Stanley depth of powers, integral closure of powers, and symbolic powers of monomial ideals.

On the law of nullity

1982 ◽  
Vol 91 (3) ◽  
pp. 357-374 ◽  
Author(s):  
P. M. Cohn ◽  
A. H. Schofield
Keyword(s):  
Global Dimension ◽  
The Other ◽  
Rank Function ◽  
Projective Modules ◽  
Natural Numbers ◽  
Finitely Generated ◽  
Skew Field ◽  
Other Hand ◽  
Semihereditary Rings

In chapter 7 of (2) conditions were given for a ring to be embeddable in a skew field; in particular, it was shown that any semifir has a universal field of fractions, over which all full matrices can be inverted. This was generalized in two different directions, by Bergman (in a letter to one of the authors in 1971) and by Dicks and Sontag(7). Dicks and Sontag characterized those rings having a field of fractions in which all full matrices are inverted; they showed that this is equivalent to Sylvester's law of nullity, and further showed that this forces the ring to have weak global dimension not exceeding 2 and all finitely generated projective modules to be free. Bergman on the other hand investigated weakly semihereditary rings having a rank function on projective modules which takes values in the natural numbers. He showed that there was a homomorphism from any such ring to a field of fractions in which every full map between finitely generated projective modules is inverted. Weakly semihereditary rings with a rank function to the natural numbers are the analogue of semifirs and so it is natural to look for a characterization of rings with a rank function on projective modules such that all full maps between projective modules become invertible in a suitable field of fractions. We shall find that, as before, this is the case if and only if Sylvester's law of nullity holds with respect to the rank function, for maps between projective modules. Further, the ring must have weak global dimension at most two. This is the content of Sections 2 and 3.

scholarly journals On the Bishop-Phelps-Bollobás Property for Numerical Radius

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Sun Kwang Kim ◽  
Han Ju Lee ◽  
Miguel Martín
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
Sufficient Conditions ◽  
The Other ◽  
Numerical Radius ◽  
Infinite Dimensional ◽  
Other Hand ◽  
Nikodym Property

We study the Bishop-Phelps-Bollobás property for numerical radius (in short, BPBp-nu) and find sufficient conditions for Banach spaces to ensure the BPBp-nu. Among other results, we show thatL1μ-spaces have this property for every measureμ. On the other hand, we show that every infinite-dimensional separable Banach space can be renormed to fail the BPBp-nu. In particular, this shows that the Radon-Nikodým property (even reflexivity) is not enough to get BPBp-nu.

scholarly journals On the adjoint group of some radical rings

1997 ◽  
Vol 39 (1) ◽  
pp. 35-41 ◽  
Author(s):  
Oliver Dickenschied
Keyword(s):  
Abelian Subgroup ◽  
Jacobson Radical ◽  
Additive Group ◽  
The Other ◽  
Adjoint Group ◽  
Radical Ring ◽  
Finiteness Conditions ◽  
Prüfer Rank ◽  
Carry Over ◽  
Minimax Condition

A ring R is called radical if it coincides with its Jacobson radical, which means that Rforms a group under the operation a ° b = a + b + ab for all a and b in R. This group is called the adjoint group R° of R. The relation between the adjoint group R° and the additive group R+ of a radical rin R is an interesting topic to study. It has been shown in [1] that the finiteness conditions “minimax”, “finite Prufer rank”, “finite abelian subgroup rank” and “finite torsionfree rank” carry over from the adjoint group to the additive group of a radical ring. The converse is true for the minimax condition, while it fails for all the other above finiteness conditions by an example due to Sysak [6] (see also [2, Theorem 6.1.2]). However, we will show that the converse holds if we restrict to the class of nil rings, i.e. the rings R such that for any a є R there exists an n = n(a) with an = 0.

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