Keir Lockridge

The ghost dimension of a ring

We introduce the concept of the ghost dimension gh.dim. R of a ring R. This is the longest nontrivial chain of maps in the derived category emanating from a perfect complex such that each map is zero on homology. We show that w.dim. R < gh.dim. R with equality if R is coherent or w.dim. R = 1.

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.

How Many Units Can a Commutative Ring Have

Abstract László Fuchs posed the following problem in 1960, which remains open: determine whether a given abelian group can occur as the group of units in a commutative ring. In this note, we provide an elementary solution to a simpler, related problem: find all cardinal numbers occurring as the cardinality of the group of units in a commutative ring. As a byproduct, we obtain a solution to Fuchss problem for the class of finite abelian p-groups when p is an odd prime.