Ideal Extensions and Directly Infinite Algebras

Abstract

Directly infinite algebras, those algebras, $E$ which have a pair of elements $x$ and $y$ where $1=xy\ne yx$, are well known to have a sub-algebra isomorphic to $M_{\infty }(K)$, the set of infinite $\zplus \times \zplus$-indexed matrices which have only finitely many nonzero entries. When this sub-algebra is actually an ideal, we may analyze the algebra in terms of an extension of some algebra $A$ by $M_{\infty }(K)$, that is, a short exact sequence of $K$-algebras $0\to M_{\infty }(K)\to E\to A\to 0$. The present article characterizes all trivial (split) extensions of $K[x,x^{-1}]$ by $M_{\infty }(K)$ by examining the extensions as sub-algebras of infinite matrix algebras. Furthermore, we construct an infinite family of pairwise non-isomorphic extensions $\{{\mathcal{T}}_i:i\ge 0\}$, all of which can be written as an extension $0\to M_{\infty }(K)\to {\mathcal{T}}_i\to K[x,x^{-1}]\to 0$.

arXiv comment19 Pages. Significant revision of the previous version. To appear in Pure and Applied Algebra