25 research outputs found
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
Sophie Germain Primes and Involutions of Znx
In the paper âWhat is special about the divisors of 24?â, 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
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
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
The generating hypothesis in the derived category of R-modules
AbstractIn this paper, we prove a version of Freydâs generating hypothesis for triangulated categories: if D is a cocomplete triangulated category and SâD is an object whose endomorphism ring is graded commutative and concentrated in degree zero, then S generates (in the sense of Freyd) the thick subcategory determined by S if and only if the endomorphism ring of S is von Neumann regular. As a corollary, we obtain that the generating hypothesis is true in the derived category of a commutative ring R if and only if R is von Neumann regular. We also investigate alternative formulations of the generating hypothesis in the derived category. Finally, we give a characterization of the Noetherian stable homotopy categories in which the generating hypothesis is true