We introduce a concept and develop a theory of Galois subalgebras in skew semigroup rings. Proposed approach has a strong impact on the representation theory, first of all the theory of Harish-Chandra modules, of many infinite dimensional algebras including the Generalized Weyl algebras, the universal enveloping algebras of reductive Lie algebras, their quantizations, Yangians etc. In particula...

In recent work of T. Cassidy and the author, a notion of complete intersection was defined for (non-commutative) regular skew polynomial rings, defining it using both algebraic and geometric tools, where the commutative definition is a special case. In this article, we extend the definition to a larger class of algebras that contains regular graded skew Clifford algebras, the coordinate ring of...

Skew lattices are the most successful generalization of lattices to the noncommutative case to date. Roughly speaking, each skew lattice can be seen as a lattice of rectangular bands. A coset decomposition can be given to each pair of comparable maximal rectangular bands. The internal structure of skew lattices is revealed by their coset structure. In the present paper we study the coset struct...

We give a new characterization of the bounded nilradical. Using the interplay between this and previous characterizations, we prove that the bounded nilradical is a graded ideal if the ring is graded. This allows us to completely determine the bounded nilradical of skew polynomial and skew Laurent polynomial rings in terms of information in the coefficient ring.

The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming distance is gained, and it is possible to design efficient decoding algorithms. In this paper, we give a version of the Hartmann-Tzeng bound that works for a wide ...