2 research outputs found
Efficient quantum processing of ideals in finite rings
Suppose we are given black-box access to a finite ring R, and a list of
generators for an ideal I in R. We show how to find an additive basis
representation for I in poly(log |R|) time. This generalizes a recent quantum
algorithm of Arvind et al. which finds a basis representation for R itself. We
then show that our algorithm is a useful primitive allowing quantum computers
to rapidly solve a wide variety of problems regarding finite rings. In
particular we show how to test whether two ideals are identical, find their
intersection, find their quotient, prove whether a given ring element belongs
to a given ideal, prove whether a given element is a unit, and if so find its
inverse, find the additive and multiplicative identities, compute the order of
an ideal, solve linear equations over rings, decide whether an ideal is
maximal, find annihilators, and test the injectivity and surjectivity of ring
homomorphisms. These problems appear to be hard classically.Comment: 5 page
Algorithms for finite rings
In this thesis we are interested in describing algorithms that answer questions arising in ring and module theory. Our focus is on deterministic polynomial-time algorithms and rings and modules that are finite.
The first main result of this thesis is a solution to the module isomorphism problem in the finite case. Further, we show how to compute a set of generators of minimal cardinality for a given finite module, and how to construct projective covers and injective hulls. We also describe tests for module simplicity, projectivity, and injectivity, and constructive tests for existence of surjective module homomorphisms between two finite modules, one of which is projective. As a negative result, we show that the problem of testing for existence of injective module homomorphisms between two finite modules, one of which is projective, is NP-complete.
The last part of the thesis is concerned with finding a good working approximation of the Jacobson radical of a finite ring, that is, a two-sided nilpotent ideal such that the corresponding quotient ring is “almost” semisimple. The notion we use to approximate semisimplicity is that of separability.ALGANTNumber theory, Algebra and Geometr