论文标题
理想的扩展和直接无限代数
Ideal Extensions and Directly Infinite Algebras
论文作者
论文摘要
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 sub-algebra isomorphic to $M_\infty(K)$, the set of infinite $\zplus \times \zplus$-indexed matrices which have only finitely many nonzero entries.当这个子代数实际上是理想的时，我们可以根据$ m_ \ infty（k）$的某些代数$ a $的扩展来分析代数，也就是说，简短的$ k $ -algebras $ 0 \ to $ k $ algebras $ 0 \ to m_ \ infty（k）\ to e \至e \至本文通过$ m_ \ infty（k）$来表征$ k [x，x^{ - 1}] $的所有微不足道（拆分）扩展，通过检查扩展名作为Infinite Matrix代数的子代数，以此。此外，我们构建了一个无限的非同构非形态扩展，$ \ {\ Mathcal t_i：i \ geq 0 \} $，所有这些都可以写为扩展名$ 0 \ to m_ \ to m_ \ infty（k）
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 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 $\{\mathcal T_i : i \geq 0\}$, all of which can be written as an extension $0 \to M_\infty(K) \to \mathcal T_i \to K[x,x^{-1}] \to 0$.
