Fuchs' problem for indecomposable abelian groups

More than 50 years ago, Laszlo Fuchs asked which abelian groups can be the group of units of a commutative ring. Though progress has been made, the question remains open. We provide an answer to this question in the case of indecomposable abelian groups by classifying the indecomposable abelian groups that are realizable as the group of units of a ring of any given characteristic.Comment: 10 pages, accepted for publication in Journal of Algebr

The generating hypothesis in the derived category of a ring

We show that a strong form (the fully faithful version) of the generating hypothesis, introduced by Freyd in algebraic topology, holds in the derived category of a ring R if and only if R is von Neumann regular. This extends results of the second author. We also characterize rings for which the original form (the faithful version) of the generating hypothesis holds in the derived category of R. These must be close to von Neumann regular in a precise sense, and, given any of a number of finiteness hypotheses, must be von Neumann regular. However, we construct an example of such a ring that is not von Neumann regular, and therefore does not satisfy the strong form of the generating hypothesis

Sophie Germain Primes and Involutions of Znx

In the paper “What is special about the divisors of 24?”, Sunil Chebolu proved an interesting result about the multiplication tables of Zn from several different number theoretic points of view: all of the 1s in the multiplication table for Zn are located on the main diagonal if and only if n is a divisor of 24. Put another way, this theorem characterizes the positive integers n with the property that the proportion of 1s on the diagonal is precisely 1. The present work is concerned with finding the positive integers n for which there is a given fixed proportion of 1s on the diagonal. For example, when p is prime, we prove that there exists a positive integer n such that 1/p of the 1s lie on the diagonal of the multiplication table for Zn if and only if p is a Sophie Germain prime

Characterizations of Mersenne and 2-rooted primes

We give several characterizations of Mersenne primes (Theorem 1.1) and of primes for which 2 is a primitive root (Theorem 1.2). These characterizations involve group algebras, circulant matrices, binomial coefficients, and bipartite graphs.Comment: 19 pages, final version, to appear in Finite Fields and their Application

DNA transposons have colonized the genome of the giant virus Pandoravirus salinus

BACKGROUND: Transposable elements are mobile DNA sequences that are widely distributed in prokaryotic and eukaryotic genomes, where they represent a major force in genome evolution. However, transposable elements have rarely been documented in viruses, and their contribution to viral genome evolution remains largely unexplored. Pandoraviruses are recently described DNA viruses with genome sizes that exceed those of some prokaryotes, rivaling parasitic eukaryotes. These large genomes appear to include substantial noncoding intergenic spaces, which provide potential locations for transposable element insertions. However, no mobile genetic elements have yet been reported in pandoravirus genomes. RESULTS: Here, we report a family of miniature inverted-repeat transposable elements (MITEs) in the Pandoravirus salinus genome, representing the first description of a virus populated with a canonical transposable element family that proliferated by transposition within the viral genome. The MITE family, which we name Submariner, includes 30 copies with all the hallmarks of MITEs: short length, terminal inverted repeats, TA target site duplication, and no coding capacity. Submariner elements show signs of transposition and are undetectable in the genome of Pandoravirus dulcis, the closest known relative Pandoravirus salinus. We identified a DNA transposon related to Submariner in the genome of Acanthamoeba castellanii, a species thought to host pandoraviruses, which contains remnants of coding sequence for a Tc1/mariner transposase. These observations suggest that the Submariner MITEs of P. salinus belong to the widespread Tc1/mariner superfamily and may have been mobilized by an amoebozoan host. Ten of the 30 MITEs in the P. salinus genome are located within coding regions of predicted genes, while others are close to genes, suggesting that these transposons may have contributed to viral genetic novelty. CONCLUSIONS: Our discovery highlights the remarkable ability of DNA transposons to colonize and shape genomes from all domains of life, as well as giant viruses. Our findings continue to blur the division between viral and cellular genomes, adhering to the emerging view that the content, dynamics, and evolution of the genomes of giant viruses do not substantially differ from those of cellular organisms. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1186/s12915-015-0145-1) contains supplementary material, which is available to authorized users