Some new results about a conjecture by Brian Alspach

In this paper, we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset A of $$\mathbb {Z}_n{\setminus } \{0\}$$ Z n \ { 0 } of size k such that $$\sum _{z\in A} z\not = 0$$ ∑ z ∈ A z ≠ 0 , it is possible to find an ordering $$(a_1,\ldots ,a_k)$$ ( a 1 , … , a k ) of the elements of A such that the partial sums $$s_i=\sum _{j=1}^i a_j$$ s i = ∑ j = 1 i a j , $$i=1,\ldots ,k$$ i = 1 , … , k , are nonzero and pairwise distinct. This conjecture is known to be true for subsets of size $$k\le 11$$ k ≤ 11 in cyclic groups of prime order. Here, we extend this result to any torsion-free abelian group and, as a consequence, we provide an asymptotic result in $$\mathbb {Z}_n$$ Z n . We also consider a related conjecture, originally proposed by Ronald Graham: given a subset A of $$\mathbb {Z}_p{\setminus }\{0\}$$ Z p \ { 0 } , where p is a prime, there exists an ordering of the elements of A such that the partial sums are all distinct. Working with the methods developed by Hicks, Ollis, and Schmitt, based on Alon’s combinatorial Nullstellensatz, we prove the validity of this conjecture for subsets A of size 12.

Neutrosophic Triplets in Neutrosophic Rings

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.

A Note on the Square Subgroups of Decomposable Torsion-Free Abelian Groups of Rank Three

A hypothesis stated in [16] is confirmed for the case of associative rings. The answers to some questions posed in the mentioned paper are also given. The square subgroup of a completely decomposable torsion-free abelian group is described (in both cases of associative and general rings). It is shown that for any such a group A, the quotient group modulo the square subgroup of A is a nil-group. Some results listed in [16] are generalized and corrected. Moreover, it is proved that for a given abelian group A, the square subgroup of A considered in the class of associative rings, is a characteristic subgroup of A.

Torsion-free abelian groups with optimal Scott families

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.

A TORSION-FREE ABELIAN GROUP EXISTS WHOSE QUOTIENT GROUP MODULO THE SQUARE SUBGROUP IS NOT A NIL-GROUP

The first example of a torsion-free abelian group $(A,+,0)$ such that the quotient group of $A$ modulo the square subgroup is not a nil-group is indicated (for both associative and general rings). In particular, the answer to the question posed by Stratton and Webb [‘Abelian groups, nil modulo a subgroup, need not have nil quotient group’, Publ. Math. Debrecen27 (1980), 127–130] is given for torsion-free groups. A new method of constructing indecomposable nil-groups of any rank from $2$ to $2^{\aleph _{0}}$ is presented. Ring multiplications on $p$-pure subgroups of the additive group of the ring of $p$-adic integers are investigated using only elementary methods.

On the Koszul map of Lie algebras

We 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.