Efficient algorithms are presented for factoring polynomials in the skew-polynomial ring F[x; σ], a non-commutative generalization of the usual ring of polynomials F[x], where F is a finite field and σ: F → F is an automorphism (iterated Frobenius map). Applications include fast functional decomposition algorithms for a class of polynomials in F[x] whose decompositions are “wild” and previously...

The first step in investigating colour symmetries for periodic and aperiodic systems is the determination of all colouring schemes that are compatible with the symmetry group of the underlying structure, or with a subgroup of it. For an important class of colourings of planar structures, this mainly combinatorial question can be addressed with methods of algebraic number theory. We present the ...

The lattice of z-ideals of the ring C(X) of real-valued continuous functions on a completely regular Hausdorff space X has been shown by Mart́ınez and Zenk to be a complete Heyting algebra with certain properties. We show that these properties are due only to the fact that C(X) is an f -ring with bounded inversion. This we do by studying lattices of algebraic z-ideals of abstract f -rings with b...

Many observations about coalgebras were inspired by comparable situations for algebras. Despite the prominent role of prime algebras, the theory of a corresponding notion for coalgebras was not well understood so far. Coalgebras C over fields may be called coprime provided the dual algebra C∗ is prime. This definition, however, is not intrinsic it strongly depends on the base ring being a field...

The classification of rings of algebraic integers which are Euclidean (not necessarily for the norm function) is a major unsolved problem. Assuming the Generalized Riemann Hypothesis, Weinberger [7] showed in 1973 that for algebraic number fields containing infinitely many units the ring of integers R is a Euclidean domain if and only if it is a principal ideal domain. Since there are principal...

Abstract By modifying constructions of Bĕıdar and Small we prove that for countably generated prime F -algebras of finite GK dimension there exists an affinization having finite GK dimension. Using this result we show: for any field there exists a prime affine algebra of GK dimension two that is neither primitive nor PI; for any countable field F there exists a prime affine F -algebra of GK dim...

We attempt to survey some of the interrelationships between algebraic K-theory, topology, and analysis. The exposition is in two parts, the first dealing with K-theory of group rings and connections with topology, and the second dealing with flat bundles, secondary invariants, and connections with differential geometry.

Let Ω be an algebraic closure of Qp and let F be a finite extension of Qp contained in Ω. Given positive integers f and e, the number of extensions K/F contained in Ω with residue degree f and ramification index e was computed by Krasner. This paper is concerned with the number I(F, f, e) of F -isomorphism classes of such extensions. We determine I(F, f, e) completely when p2 ∤ e and get partia...