scholarly journals Note on U-closed semigroup rings

1999 ◽  
Vol 59 (3) ◽  
pp. 467-471 ◽  
Author(s):  
Ryûki Matsuda
Keyword(s):  
Abelian Group ◽  
Integral Domain ◽  
Free Abelian Group ◽  
Semigroup Ring ◽  
Closed Domain ◽  
Semigroup Rings ◽  
Quotient Field ◽  
Torsion Free Abelian Group ◽  
Torsion Free

Let D be an integral domain with quotient field K. If α2 − α ∈ D and α3 − α2 ∈ D imply α ∈ D for all elements α of K, then D is called a u-closed domain. A submonoid S of a torsion-free Abelian group is called a grading monoid. We consider the semigroup ring D[S] of a grading monoid S over a domain D. The main aim of this note is to determine conditions for D[S] to be u-closed. We shall show the following Theorem: D[S] is u-closed if and only if D is u-closed.

scholarly journals Neutrosophic Triplets in Neutrosophic Rings

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 563 ◽  
Author(s):  
Vasantha Kandasamy W. B. ◽  
Ilanthenral Kandasamy ◽  
Florentin Smarandache
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Torsion Free Abelian Group ◽  
Ring Of Integers ◽  
Torsion Free

The neutrosophic triplets in neutrosophic rings ⟨ Q ∪ I ⟩ and ⟨ R ∪ I ⟩ are investigated in this paper. However, non-trivial neutrosophic triplets are not found in ⟨ Z ∪ I ⟩ . In the neutrosophic ring of integers Z ∖ { 0 , 1 } , no element has inverse in Z. It is proved that these rings can contain only three types of neutrosophic triplets, these collections are distinct, and these collections form a torsion free abelian group as triplets under component wise product. However, these collections are not even closed under component wise addition.

scholarly journals On Crossed Products of a Sfield

1963 ◽  
Vol 22 ◽  
pp. 65-71 ◽  
Author(s):  
Masatoshi Ikeda
Keyword(s):  
Integral Domain ◽  
Free Abelian Group ◽  
Group Extension ◽  
Crossed Products ◽  
Torsion Free Abelian Group ◽  
Outer Automorphisms ◽  
Identity Automorphism ◽  
Limit Ordinal ◽  
Group B ◽  
Torsion Free

In the previous paper [3] the author has shown a possibility to construct a series of sfields by taking sfields of quotients of split crossed products of a sfield. In this paper the same problem is treated, and, by considering general crossed products, a generalization of the previous result is given: Let K be a sfield and G be the join of a well-ordered ascending chain of groups Gα of outer automorphisms of K such that a) G1 is the identity automorphism group, b) Gα is a group extension of Gα-1 by a torsion-free abelian group for each non-limit ordinal α, and c) for each limit ordinal α. Then an arbitrary crossed product of K with G is an integral domain with a sfield of quotients Q and the commutor ring of K in Q coincides with the centre of K.

scholarly journals Type radicals and quasi-decompositions of torsion-free abelian groups

1992 ◽  
Vol 52 (2) ◽  
pp. 219-236 ◽  
Author(s):  
Oteo Mutzbauer
Keyword(s):  
Abelian Group ◽  
Finite Rank ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractA composition sequence for a torsion-free abelian group A is an increasing sequenceof pure subgroups with rank 1 quotients and union A. Properties of A can be described by the sequence of types of these quotients. For example, if A is uniform, that is all the types in some sequence are equal, then A is complete decomposable if it is homogeneous. If A has finite rank and all permutations ofone of its type sequences can be realized, then A is quasi-isomorphic to a direct sum of uniform groups.

A THEOREM ON MATRIX GROUPS

2008 ◽  
Vol 18 (01) ◽  
pp. 165-180
Author(s):  
A. I. PAPISTAS
Keyword(s):  
Abelian Group ◽  
Finite Subset ◽  
Free Abelian Group ◽  
Principal Ideal Domain ◽  
Matrix Groups ◽  
Generating Set ◽  
Torsion Free Abelian Group ◽  
Group Of Units ◽  
Multiplicative Monoid ◽  
Torsion Free

Let K be a principal ideal domain, and An, with n ≥ 3, be a finitely generated torsion-free abelian group of rank n. Let Ω be a finite subset of KAn\{0} and U(KAn) the group of units of KAn. For a multiplicative monoid P generated by U(KAn) and Ω, we prove that any generating set for [Formula: see text] contains infinitely many elements not in [Formula: see text]. Furthermore, we present a way of constructing elements of [Formula: see text] not in [Formula: see text] for n ≥ 3. In the case where K is not a field the aforementioned results hold for n ≥ 2.

Torsion-free abelian groups with optimal Scott families

2018 ◽  
Vol 18 (01) ◽  
pp. 1850002
Author(s):  
Alexander G. Melnikov
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Symbolic Logic ◽  
Technical Result ◽  
Torsion Free Abelian Group ◽  
Scott Family ◽  
Free Abelian Groups ◽  
Torsion Free ◽  
Labeled Tree

We prove that for any computable successor ordinal of the form [Formula: see text] [Formula: see text] limit and [Formula: see text] there exists computable torsion-free abelian group [Formula: see text]TFAG[Formula: see text] that is relatively [Formula: see text] -categorical and not [Formula: see text] -categorical. Equivalently, for any such [Formula: see text] there exists a computable TFAG whose initial segments are uniformly described by [Formula: see text] infinitary computable formulae up to automorphism (i.e. it has a c.e. uniformly [Formula: see text]-Scott family), and there is no syntactically simpler (c.e.) family of formulae that would capture these orbits. As far as we know, the problem of finding such optimal examples of (relatively) [Formula: see text]-categorical TFAGs for arbitrarily large [Formula: see text] was first raised by Goncharov at least 10 years ago, but it has resisted solution (see e.g. Problem 7.1 in survey [Computable abelian groups, Bull. Symbolic Logic 20(3) (2014) 315–356]). As a byproduct of the proof, we introduce an effective functor that transforms a [Formula: see text]-computable worthy labeled tree (to be defined) into a computable TFAG. We expect that this technical result will find further applications not necessarily related to categoricity questions.

scholarly journals Hypertypes of torsion-free abelian groups of finite rank

1989 ◽  
Vol 39 (1) ◽  
pp. 21-24 ◽  
Author(s):  
H.P. Goeters ◽  
C. Vinsonhaler ◽  
W. Wickless
Keyword(s):  
Abelian Group ◽  
Finite Rank ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Free Subgroup ◽  
Torsion Free Abelian Group ◽  
Free Abelian Groups ◽  
Torsion Free

Let G be a torsion-free abelian group of finite rank n and let F be a full free subgroup of G. Then G/F is isomorphic to T1 ⊕ … ⊕ Tn, where T1 ⊆ T2 ⊆ … ⊆ Tn ⊆ ℚ/ℤ. It is well known that type T1 = inner type G and type Tn = outer type G. In this note we give two characterisations of type Ti for 1 < i < n.

scholarly journals On the Koszul map of Lie algebras

2016 ◽  
Vol 28 (1) ◽  
Author(s):  
Yves Cornulier
Keyword(s):  
Abelian Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Free Abelian Group ◽  
Bilinear Maps ◽  
Nilpotent Lie Algebra ◽  
Torsion Free Abelian Group ◽  
The Third ◽  
Torsion Free

AbstractWe motivate and study the reduced Koszul map, relating the invariant bilinear maps on a Lie algebra and the third homology. We show that it is concentrated in degree 0 for any grading in a torsion-free abelian group, and in particular it vanishes whenever the Lie algebra admits a positive grading. We also provide an example of a 12-dimensional nilpotent Lie algebra whose reduced Koszul map does not vanish. In an appendix, we reinterpret the results of Neeb and Wagemann about the second homology of current Lie algebras, which are closely related to the reduced Koszul map.

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