Skew Polynomials and Algebraic Reflexivity

Abstract

Abstract. For an arbitrary K-algebra R, an R, K-bimodule M is algebraically reflexive if the only K-endomorphisms of M leaving invariant every R-submodule of M are the scalar multiplications by elements of R. Hadwin has shown for an infinite field K and R = K[x] that R is reflexive as an R, K-bimodule. This paper provides a generalisation by giving a skew polynomial version of his result.2000 Mathematics Subject Classification. Primary 16D20, 16S36. Introduction.If V is a vector space over a field K and L is a lattice of subspaces then alg L is defined to be the algebra of all K-endomorphisms of V which leave L point-wise fixed; dually if R is a subalgebra of End K V , then lat R is defined to be the lattice of all subspaces of V which are left invariant by every element of R. Combining these two functors produces alg (lat R), an algebra containing R, and R is called reflexive when these two algebras are equal. Thus, an algebra of operators on a vector space is called reflexive when no larger algebra of operators has the same lattice of invariant subspaces. This terminology originated with Halmos [3] although earlier work also explored some of this area; for example, the lattice of invariant subspaces of a single linear transformation was studied in [1].In [2] a start was made on studying an extension of the concept of alg lat where the focus moved to classes of representations, that is R, K-bimodules, of a K-algebra R. For an arbitrary K-algebra R a bimodule R M K is said to be (algebraically) reflexive if, whenever α ∈ End K M is such that αm ∈ Rm for all m ∈ M, then there is an r ∈ R such that α is simply left multiplication by r.More generally, define for rings R, A and bimodule R M A [2]