Given a field K, a Kronecker module V is a pair of K-linear spaces (U, V ) together with a K-bilinear map K 2 ร U โ V . In finite dimensions this is also the notion of pencils of matrices. Every K[X]-module can be construed as a Kronecker module. In particular the K[X]-submodules of K(X) give rise to the Kronecker modules R h where h is a height function, i.e. a function h : K โช {โ} โ {โ, 0, 1, 2, . . .}. The K[X]-module K[X] itself gives the Kronecker module P that goes with the height function which is โ at โ and 0 on K. The modules R h that are infinite-dimensional come up precisely when h attains the value โ or when h is stictly positive on an infinite subset of K โช {โ}. The endomorphism algebra of R h is called a pole algebra. Those Kronecker modules V that are extensions of finite-dimensional submodules of P by infinite-dimensional R h lead to some engaging problems with matrix algebras. This is because the endomorphisms of such V constitute a K-subalgebra of n ร n matrices over K(X), which is commutative if the extension is indecomposable. Among the algebras that are known to arise in the 2 ร 2 case, when the extension is by P, are the coordinate rings of all elliptic curves. In this paper we replace P by an arbitrary infinite-dimensional R h . The following new algebras are realized: infinite-dimensional pole algebras End R h where h = 0 on an infinite subset of K; maximal subalgberas of A ร B for some pole algebras A, B; the quasi local ring K โ S where S is a K-vector space of dimension at most card K. In the process we identify those height functions h that will tolerate an indecomposable extension V having non-trivial endomorphisms.