Ideal Extensions and Directly Infinite Algebras
Abstract
Directly infinite algebras, those algebras, $E$ which have a pair of elements $x$ and $y$ where $1 = xy \neq yx$, are well known to have a subalgebra isomorphic to $M_\infty(K)$, the set of infinite $\zplus \times \zplus$indexed matrices which have only finitely many nonzero entries. When this subalgebra 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 subalgebras of infinite matrix algebras. Furthermore, we construct an infinite family of pairwise nonisomorphic extensions $\{\mathcal T_i : i \geq 0\}$, all of which can be written as an extension $0 \to M_\infty(K) \to \mathcal T_i \to K[x,x^{1}] \to 0$.
 Publication:

arXiv eprints
 Pub Date:
 September 2020
 arXiv:
 arXiv:2009.04055
 Bibcode:
 2020arXiv200904055B
 Keywords:

 Mathematics  Rings and Algebras
 EPrint:
 19 Pages. Significant revision of the previous version. To appear in Pure and Applied Algebra