36 citations · 90 across the 10 of their papers we have counts for
5 papers · 1 filter
Realizing orders as group rings
H. W. Lenstra, A. Silverberg, D. M. H. van Gent
An order is a commutative ring that as an abelian group is finitely generated and free. A commutative ring is reduced if it has no non-zero nilpotent elements. In this paper we use…
Universal gradings of orders
H. W. Lenstra, A. Silverberg
For commutative rings, we introduce the notion of a {\em universal grading}, which can be viewed as the "largest possible grading". While not every commutative ring (or order) has…
Algorithms for commutative algebras over the rational numbers
H. W. Lenstra, A. Silverberg
The algebras considered in this paper are commutative rings of which the additive group is a finite-dimensional vector space over the field of rational numbers. We present determin…
Roots of unity in orders
H. W. Lenstra, A. Silverberg
We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the or…
Determining cyclicity of finite modules
H. W. Lenstra, A. Silverberg
We present a deterministic polynomial-time algorithm that determines whether a finite module over a finite commutative ring is cyclic, and if it is, outputs a generator.