Polynomials, series and Kronecker modules

Abstract

A Kronecker module over a field K may be viewed as a quadruple ( V, W; a, b), with a, b K-linear maps from space V to space W. For an indeterminate X let P be the module ( K[ X], K[ X]; identity, multiplication by X). All purely simple Kronecker modules of rank-2, with P as homomorphic image, may be indexed by formal power series α. Homomorphisms and isomorphisms among such P α's are studied by means of operations on polynomials and formal power series. For instance, if P α ≊ P β then ( rα + s) β = tα + u, for some polynomials r, s, t, u not all 0. Also for those α's that are not roots of a linear or quadratic equation over K[ X], End P α = K.