Ideal extensions and directly infinite algebras

Directly infinite algebras, those E which have a pair of elements x and y where 1=xy≠yx, are well known to sub-algebra isomorphic M∞(K), the set Z+×Z+-indexed matrices only finitely many nonzero entries. When this is actually an ideal, we may analyze algebra in terms extension some A by that is, short exact sequence K-algebras 0→M∞(K)→E→A→0. The present article characterizes all trivial (split) extensions K[x,x−1] M∞(K) examining as sub-algebras matrix algebras. Furthermore, construct family pairwise non-isomorphic {Ti:i≥0}, can be written 0→M∞(K)→Ti→K[x,x−1]→0.